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NCERT exemplar Class 12 Maths solutions chapter 11 Three Dimensional Geometry - If you observe the world around you, you see everything in three dimensions! Even a tiny strand of hair has dimensions of length, width, and depth. 3D geometry refers to the mathematics of perception, direction, and shape. NCERT exemplar Class 12 Maths chapter 11 solutions initiates that there is always a requirement of three parameters to work with the comprehensive concepts of three-dimensional geometry. NCERT exemplar solutions For Class 12 Maths chapter 11 would help you out in any three-dimensional Mathematical problem that you're stuck on and get you back in your study loop. NCERT exemplar Class 12 Maths solutions chapter 11 PDF download is helpful for students to learn offline when there is a slow internet connection. Also, read - NCERT Class 12 Maths Solutions
Question:1
Answer:
Given, is inclined at 600 to and at 450 to
= 10 units.
We want to find the position vector of point A in space, which is nothing but
We know, there are three axes in space: X, Y, and Z.
Let OA be inclined with OZ at an angle α.
We know, directions cosines are associated by the relation:
l² + m² + n² = 1 ….(i)
In this question, direction cosines are the cosines of the angles inclined by on , , and
So,
Substituting the values of l, m, and n in equation (i),
We know the values of and , i.e. 1/2 and 1/√2 respectively.
Therefore, we get
So is given as
..........(ii)
We have,
Inserting these values of l, m and n in equation (ii),
Also ,Put
Thus, position vector of A in space
Question:2
Answer:
Given, vector =
Point = (1, -2, 3)
We can write this point in vector form as
Let ,
We must find the vector equation of the line parallel to the vector and passing through the point
We know, equation of a line passing through a point and parallel to a given vector is denoted as
Where,
In other words, we need to find
This can be achieved by substituting the values of the vectors in the above equation. We get
This can be further rearranged, upon which we get:
Thus, the require vector equation of line is
which can also be written as
Question:3
Show that the given lines, and intersect.
Also, find the point of intersection of the lines.
Answer:
We have the lines,
Let us denote these lines as L1and L2, such that
where
We must show that the lines L1and L2 intersect.
To show this, let us first find any point on line L1 and line L2
For L1:
We must find the values of x, y, and z. Therefore, let us take
Take
Take
Therefore, any point on L1 can be represented as .
Now,
For L2:
We must find the values of x, y, and z. Therefore,
Take
Take
Take
Hence, any point on line L? can be represented as (5μ + 4, 2μ + 1, μ).
If lines L1 and L2 intersect, then there exist λ and μ such that
Substituting the value of μ from equation (v) into equation (iv),
Putting this value of in eq (v),
To check, we can substitute the values of and in equation (iii), giving us:
Therefore and also satisfy equation (iii).
So, the z-coordinate from equation (i),
And the z-coordinate from equation (ii),
So, the lines intersect at the point
Therefore the lines intersect at the point (-1, -1, -1).
Question:4
Find the angle between the lines and
Answer:
Given, lines:
We are instructed to find the angle between the lines.
The line is parallel to the vector
Let
Then, we can say the line is parallel to vector
Similarly, let
Then, we can say is parallel to the vector
If we take θ as the angle between the lines, then cosine θ is:
Substituting the values of and in the above equation,
We get
Here,
Also,
Substituting the values of in equation (i) and (ii), we get
Therefore, the angle between the lines is
Question:5
Answer:
Given: A (0, -1, -1), B (4, 5, 1), C (3, 9, 4), D (-4, 4, 4).
To prove: The line passing through A and B intersects the line passing through C and D.
Proof: We know, equation of a line passing through two points (x1 , y1 , z1) and (x2 , y2 , z2) is:
Hence, the equation of the line passing through A (0, -1, -1) and B (4, 5,1) is:
, where x1 = 0, y1 = -1, z1 = -1; and x2 = 4, y2 = 5, z2 = 1
Let
We must find the values of x, y, and z. Therefore,
This implies that any point on the line L1 is (4λ, 6λ – 1, 2λ – 1).
The equation of the line passing through points C (3, 9, 4) and D (-4, 4, 4) is:
, where x1 = 3, y1 = 9, z1 = 4; and x2 = -4, y2 = 4, z2 = 4
Let
We must find the values of x, y, and z. Therefore,
This implies that any point on line L2 is (-7μ +3, -5μ + 9, 4).
If the lines intersect, then there must exist a value of λ and for μ, for which
From equation (iii), we get
Substituting the value of λ in equation (i),
Substituting these values of λ and μ in equation (ii),
Since the values of λ and μ satisfy eq (ii), the lines intersect.
Hence, proved that the line through A and B intersects the line through C and D.
Question:6
Answer:
Given: x = py + q, z = ry + s, and x = p’y + q’, z = r’y + s’ are perpendicular.
To Prove: pp’ + rr’ + 1 = 0.
Proof:
Let us take x = py + q and z = ry + s.
From x = py + q;
py = x - q
From z = ry + s;
ry = z - s
So,
Or, … (i)
Now, if we take x = p’y + q’ and z = r’y + s’
From x = p’y + q’;
p’y = x - q’
From z = r’y + s’;
r’y = z - s’
So,
Or,
From (i),
Line L1 is parallel to (from the denominators of the equation (i))
From (ii),
Line L2 is parallel to (from the denominators of the equation (ii))
According to the question, L1 and L2 are perpendicular.
Therefore, the dot product of the vectors should equate to 0.
Or,
(since, in vector dot product,
Or,
Therefore, the lines are perpendicular if pp’ + rr’ + 1 = 0.
Question:7
Answer:
Given, there exists a plane which perpendicularly bisects the line joining A (2, 3, 4) and B (4, 5, 8) at right angles. We must find the equation of this plane.
First, let us find the midpoint of AB.
Since the midpoint of any line is halfway between the two end points,
= (3, 4, 6).
We can represent this as a position vector,
Next, we must find the normal of the plane,
We know, the equation of the plane which perpendicularly bisects the line joining two given points is
Where,
Substituting the values in the above equation,
Upon further simplification,
Therefore, the required equation of the plane is x + y + 2z = 19.
Question:8
Answer:
Given, the plane is at a distance of from the origin, and the normal is equally inclined to coordinate axes.
We need to find the equation of this plane.
We know, the vector equation of a plane located at a distance d from the origin is represented by:
lx + my + nz = d ….(i) , where l, m and n are the direction cosines of the normal of the plane.
Since the normal is equally inclined to the coordinate axes,
Also, we know,
This means,
if we substitute the values of l, m and n in equation (i),
So,
Therefore, the required equation of the plane is x + y + z = 9.
Question:9
Answer:
Given: the line drawn from point (-2, -1, -3) meets a plane at 900 at the point (1, -3, 3). We must find the equation of the plane.
Any line perpendicular to the plane is the normal.
Let the points be P (-2, -1, -3) and Q (1, -3, 3), then the line PQ is a normal to the plane.
Hence, PQ = (1 + 2, -3 + 1, 3 + 3)=> PQ = (3, -2, 6)
=> Normal to the plane =
The vector equation of a plane is represented by
Putting the obtained values in this equation,
We get,
Therefore, the required equation of the plane is 3x - 2y + 6z = 27.
Question:10
Find the equation of the plane through the points (2, 1, 0), (3, -2, -2) and (3, 1, 7).
Answer:
Given points are (2, 1, 0), (3, -2, -2), and (3, 1, 7).
We know, equation of a line passing through 3 non-collinear points (x1 , y1 , z1 ), (x2 , y2 , z2 ) and (x3 , y3 , z3 ) is given as:
Where, (x1 , y1 , z1 ) = (2, 1, 0)
(x2 , y2 , z2 ) = (3, -2, -2)
(x3 , y3 , z3 ) = (3, 1, 7)
Therefore, x1 = 2, y1 = 1, z1 = 0; x2 = 3, y2 = -2, z2 = -2; x3 = 3, y3 = 1, z3 = 7
Substituting these values in the line equation,
Now, since
Hence, the required equation of the plane is 7x + 3y - z = 17.
Question:11
Find the equations of the 2 lines through the origin which intersect the line at angles of each.
Answer:
Given the equation of the line, we need to find the equations of two lines through the origin which intersect the given line.
According to the theorem, equation of a line with direction ratios d1 = (b1 , b2 , b3 ) that passes through the point (x1 , y1 , z1 ) is expressed as:
We also know, the angle between two lines with direction ratios d1 and d2 respectively is given by:
We use these theorems to find the equations of the two lines.
Let the equation of a line be:
Given that it passes through the origin, (0, 0, 0)
Therefore, equation of both lines passing through the origin will be :
Let,
Direction ratio of the line = (2, 1, 1)
If we represent the direction ratio in terms of a position vector,
Any point on the line is given by (x, y, z). From (ii),
Hence, any point on line (ii) is
Since line (i) passes through the origin, we can say
We can represent the direction ratio in terms of position vector like:
From the theorem, we know
If we substitute the values of d? and d? from (iv) and (vi) in the above equation, and putting from the question:
Solving the numerator,
Solving the denominator,
And cos π/3 = 1/2
Substituting the values, we get
Performing cross multiplication,
Squaring both sides,
Therefore, from equation (v)
Direction ratio =
Putting μ = -1:
Direction Ratio = (2(-1) + 3, (-1) + 3, -1)
⇒ Direction Ratio = (-2 + 3, -1 + 3, -1)
⇒ Direction Ratio = (1, 2, -1) …(vi)
Now putting μ = -2:
Direction Ratio = (2(-2) + 3, (-2) + 3, -2)
⇒ Direction Ratio = (-4 + 3, -2 + 3, -2)
⇒ Direction Ratio = (-1, 1, -2) …(vii)
Using the direction ratios in (vi) and (vii) in equation (i);
And,
Therefore, the two required lines are and
Question:12
Answer:
Given, two lines whose direction cosines are l + m + n = 0 - (i); and l² + m² - n² = 0 - (ii). We need to find the angle between these lines.
First, we must find the values of l, m and n.
From equation (i), l + m + n = 0
=> n = - l - m
=> n = -(l + m) …(iii)
If we substitute the value of n from (i) in (ii),
⇒ l = 0 or m = 0
Putting l = 0 in equation (i),
=> 0 + m + n = 0
=> m + n = 0
=> m = -n
If m = , then
n = -m = -
Hence, direction ratios (l, m, n) = (0, , -)
=> Position vector parallel to these given lines =
Now, putting m = 0 in equation (i),
=> l + 0 + n = 0
=> l + n = 0
=> l = -n
If n = , then
l = -n = -
Hence, direction ratios (l, m, n) = (-, 0, )
=> Position vector parallel to these given lines =
From the theorem, we get the angle between the two lines whose direction ratios are d1 and d2 as:
If we substitute the values of d1 and d2, we get
Solving the numerator,
Solving the denominator,
Substituting the values in θ,
Therefore, the required angle between the lines is π/3.
Question:13
Answer:
Given: direction cosines of a variable line in two adjacent positions are l, m, n and
We have to prove that the small angle between the two positions is given by
We know, the relationships between direction cosines is given as
Also,
Let
We know, angle between two lines =
Here, the angle is very small because the line is variable in different although adjacent positions. According to the question, this small angle is
Therefore,
Substituting the values of the two vectors, we get
The dot product of 2 vectors is calculated by obtaining the sum of the product of the coefficients of
Or,
We know,
On the left-hand side, the angle is . On the right hand side, it becomes half, that is, .
Similarly replacing by in LHS, then making the angle on the RHS half,
We get:
Since is a very small angle, will be much smaller. Hence will also be very small in value.
Hence, proved.
Question:14
Answer:
We have the points O (0, 0, 0) and A (a, b, c) where a, b, and c are direction ratios. We need to find the direction cosines of line OA and the equation of the plane through A at right angle to OA.
To begin with,
We know, if (a, b, c) are the direction ratios of a given vector, then its direction cosines will be:
According to the question, the direction ratios are (a, b, c), therefore the direction cosines of the vector OA are the same as the above formula, that is,
Given, the plane is perpendicular to OA. We know, a normal is a line or vector which is perpendicular to a given object. Therefore, we can say:
Also, the vector equation of a plane where the normal is passing through the plane and passing through is,
Where
Here, the given point in the plane is A (a, b, c).
Substituting the vectors respectively, we get:
Upon simplifying this, we get:
Hence, the required equation of the plane is a² + b² + c² = ax + by + cz.
Question:15
Answer:
Given, we have 2 systems of rectangular axes. Both the systems have the same origin, and there is a plane that cuts both systems.
One system is cut at a distance of a, b, c.
The other system is cut at a distance of a’, b’, c’.
To prove:
Proof: Since a plane intersects both the systems at distances a, b, c, and a’, b’, c’ respectively, this plane will have different equations in the two different systems.
Let us consider the equation of the plane in the system with distances a, b, c to be:
Let us consider the equation of the plane in the system with distances a’, b’, c’ be:
According to the question, the plane cuts both the systems from the origin. We know, the perpendicular distance of a plane from the origin is given by:
(where not all a, b, and c are zero)
Therefore, the perpendicular distance from the origin of the first plane is:
And, the perpendicular distance from the origin of the second plane:
We also know, if two systems of lines have the same origin, their perpendicular distances from the origin to the plane in both systems are equal.
Therefore,
Cross-multiplying,
Squaring both sides,
Or
Hence, proved.
Question:16
Answer:
Given, the perpendicular from the point (let) C (2, 3, -8) to the line of which the equation is,
This can be re-written as,
Hence, the vector equation of the line is,
We must find the foot of the perpendicular from the point C (2, 3, -8) to given line, as well as the perpendicular distance from the given point C to the line.
To start with, let us locate the point of intersection between the point and the line.
Let us take,
We have,
Therefore, the coordinates of any point on the given line is
Let us consider the foot of the perpendicular from C(2, 3, -8) on line to be
Therefore, the direction ratios of
Also, the direction ratio of the line is, (-2, 6, -3).
Since L is the foot of the perpendicular on the line,
Sum of the product of these direction ratios and (-2, 6, -3) = 0.
If we substitute this value of λ in , we get
Now, we must calculate the perpendicular distance of point C from the line, that is point L.
In other words, we need to find
We know,
Substituting λ = 1,
To find
Therefore, the foot of the perpendicular from the point C to the given line is (2, 6, -2) and the perpendicular distance is units.
Question:17
Find the distance of a point (2, 4, -1) from the line
Answer:
Given, the point P (2, 4, -1), the equation of the line is
We must find the distance of point P from this line.
Note, to find the distance between a point and a line, we should get foot of the perpendicular from the point on the line.
Let, P(2, 4, -1) be the given point and be the given line.
Direction ratio of the line L is (1, 4, -9) …(i)
Let us find any point on this line.
Taking L,
Therefore, any point on the line L is
Let this point be , the foot of the perpendicular from the point P (2, 4, -1) on the line L.
Hence, the direction ratio of PQ is given by
=> Direction ratio of PQ= …(ii)
Also, we know, if two lines are perpendicular to each other, then the dot product of their direction ratios should be 0.
Here, PQ is perpendicular to L. We have, from (i) and (ii),
Direction ratio of L = (1, 4, -9)
Direction ratio of PQ =
Therefore,
Hence, the coordinate of Q, i.e. the foot of the perpendicular from the point on the given line is,
Now, to find the perpendicular distance from P to the line, that is point Q,
That is, to find
We know,
Substituting
Now, to find
Therefore, the distance from the given point to the given line = 7 units.
Question:18
Answer:
Given, point P (1, 3/2, 2)
The plane is 2x - 2y + 4z + 5 = 0
We must find the foot of the perpendicular from the point P to the equation of the given plane.
Also, we must find the distance from the point P to the plane.
Let us consider the foot of the perpendicular from point P to be Q.
Let Q be Q (x1 , y1 , z1)
So, the direction ratio of PQ is given by
(x1 - 1, y1 - 3/2, z1 - 2)
Now, let us consider the normal to the plane 2x - 2y + 4z + 5 = 0:
It is obviously parallel to PQ, since a normal is a line or vector that is perpendicular to a given object. The direction ratio simply states the number of units to move along each axis.
For any plane, ax + by + cz = d, where, a, b, and c are normal vectors to the plane.
Hence, the direction ratios are (a, b, c).
Therefore, the direction ratio of the normal = (2, -2, 4) for plane 2x - 2y + 4z + 5 = 0.
The Cartesian equation of the line PQ, where P(1, 3/2, 2) and Q (x1 , y1 , z1) is:
To find any point on this line,
Any point on the line is
This point is Q.
And, it was assumed that is lies on the given plane. Substituting x1, y1, and z1 in the plane equation, we get:
2x1 - 2y1 + 4z1 + 5 = 0
Simplifying to find the value of
Since Q is the foot of the perpendicular from the point P,
We substitute the value of in equation (i) to get:
Then, to find
Where, P = (1, 3/2, 2) and Q = (0, 5/2, 0)
Thus, the foot of the perpendicular from the given point to the plane is (0, 5/2, 0) and the distance is units.
Question:19
Answer:
Given, a line passes through a point P (3, 0, 1) and is parallel to the planes x + 2y = 0 and 3y - z = 0.
We must find the equation of this line.
Let the position vector of point P be
Or,
Let us consider the normal to the given planes, that is, perpendicular to the normal of the plane x + 2y = 0 and 3y - z = 0
Normal to the plane x + 2y = 0 can be given as
Normal to the plane 3y - z = 0 can be given as
So, is perpendicular to both these normals.
So,
Taking the 1st row and the 1st column, we multiply the 1st element of the row with the difference of products of the opposite elements , excluding 1st row and 1st column
Here,
Now, we take the 2nd column and 1st row, and multiply the 2nd element of the row (a??) with the difference of the product of opposite elements
Here
Finally, taking the 1st row and 3rd column , we multiply the 3rd element of the row (a??) with the difference of the product of opposite elements excluding the 1st row and 3rd column.
Here
Futher simplifying it,
Therefore, the direction ratio is (-2, 1, 3) …(iii)
We know, vector equation of any line passing through a point and parallel to a vector is where
Hence, from (i) and (ii),
Putting these vectors in the equation
We get
But we know,
Substituting this,
Thus, the required equation of the line is
Question:20
Answer:
Given, a plane passes through the points (2, 1, -1) and (-1, 3, 4) and is perpendicular to the plane x - 2y + 4z = 10.
We want to find the equation of this plane.
We know, the Cartesian equation of a plane passing through (x1, y1, z1)
with direction ratios perpendicular to a, b, c for its normal is given as:
a (x - x1) +b (y - y1) + c (z - z1) = 0
Hence,
Let us consider the equation of the plane passing through (2, 1, -1) to be
a(x – 2) + b(y – 1) + c(z – (-1)) = 0
⇒ a(x – 2) + b(y – 1) + c(z + 1) = 0 …(i)
Since it also passes through point (-1, 3, 4) we just replace x, y, z by -1, 3, and 4 respectively.
⇒ a(-1 – 2) + b(3 – 1) + c(4 + 1) = 0
⇒ -3a + 2b + 5c = 0 …(ii)
Since a, b, and c are direction ratios and this plane is perpendicular to the plane x - 2y + 4z = 10, we just replace x, y, and z with a, b, and c respectively (neglecting 10) and we equate this to 0.
=> a - 2b + 4c = 0 …(iii)
To solve two equations x1a + y1b + z1c = 0 and x2a + y2b + z2c = 0, we use the formula
Similarly, to solve for equations (ii) and (iii):
That is,
Substituting these values of a, b, and c in equation (i), we get
Therefore, the required equation of the plane is 18x + 17y + 4z = 49.
Question:21
Find the shortest distance between the lines given by and
Answer:
Given two lines,
Taking equation (i),
We know, the vector equation of a line passing through a point and parallel to a vector is where
= Position vector of the point through which line passes
= Normal to the line
Comparing this with equation (iii), we get
Now take equation (ii)
Similarly from (iv)
So, the shortest distance between two lines can be represented as:
solve
Taking 1st row and 1st column, we multiply the 1st element of the row (a??) with the difference of the product of the opposite elements , excluding the 1st row and the 1st column;
Here
Now, we take the 2nd column and 1st row, and multiply the 2nd element of the row (a??) with the difference of the product of opposite elements (a?? x a?? - a?? x a??)
Here
Finally, taking the 1st row and 3rd column , we multiply the 3rd element of the row (a??) with the difference of the product of opposite elements (a?? x a?? - a?? x a??), excluding the 1st row and 3rd column.
Here
Further simplifying it.
And,
Substituting the values from (v), (vi) and (vii) in d, we get
Thus, the shortest distance between the lines is 14 units.
Question:22
Answer:
Given, a plane is perpendicular to another plane 5x + 3y + 6z + 8 = 0,and also contains line of intersection of the planes x + 2y + 3z - 4 = 0 and 2x + y -z + 5 = 0.
We must find the equation of this plane.
We know, the equation of a plane passing through the line of intersection of the planes a1x + b1y + c1z + d1 = 0 and a2x + b2y + c2z + d2 = 0 is given as,
Similarly, the equation of a plane through the line of intersection of the planes x + 2y + 3z - 4 = 0 and 2x + y -z + 5 = 0. is given by,
Thus, the direction ratio of plane in (i) is,
Since the plane in equation (i) is perpendicular to the plane 5x + 3y + 6z + 8 = 0;
we can replace x, y, z with (1 + 2λ), (2 + λ) and (3 - λ) respectively in the plane 5x + 3y + 6z + 8 = 0 (neglecting 8) and equating to 0.
This gives us,
Substituting this value of in equation (i) we get
Thus, the required equation of the plane is 51x + 15y - 50z + 173 = 0.
Question:23
Answer:
Given, the plane ax + by = 0 is rotated about its line of intersection with z = 0 by an angle
To prove: equation of the plane in its new position is
Proof: Two planes are given, ax + by = 0 …(i) and z = 0 …(ii)
We know, the equation of the plane passing through the line of intersection of the planes (i) and (ii) is
where,
The angle between the new plane and plane (i) is given as
Since the angle between two planes is equivalent to the angle between their normals, the direction ratio of normal to ax + by = 0 or ax + by +0z = 0 is (a, b, 0).
And, the direction ratio of normal to is (a, b, λ).
Also, we know, angle between 2 normal vectors of the two given planes can be given as;
If we substitute the values of these vectors, we get
We then multiply by the numerator and denominator on the right hand side of the equation to get
Applying square on both sides,
Substituting the value of in equation (iii) to find the plane equation,
ax + by + λz = 0
Hence proved.
Question:24
Answer:
Given two planes,
Also given, the perpendicular distance of the plane from the origin = 1.
We must find the equation of this plane.
We know,
Simplifying the planes,
Also, for
The equation of a plane through the line of intersection of x + 3y - 6 = 0 and 3x - y - 4z = 0 can be given as
Also, we know, the perpendicular distance of a plane, ax + by + cz + d = 0 from the origin, let’s say P, is given by
Similarly, the perpendicular distance of the plane in equation (iii) from the origin (=1 according to the question) is:
Taking the square of both sides,
First, we subsitute in eq (iii) to find the plane equation
Now, we substitute λ= -1 in eq (iii) to find the plane equation
Therefore, the equation of the required plane is -2x + 4y + 4z – 6 = 0 and 4x – 2y – 4z – 6 = 0.
Question:25
Show that the points and are equidistant from the plane and lies on the opposite of it.
Answer:
Given two points,
Also,
Where,
Therefore,
We must show that the points A and B are equidistant from the plane
5x + 2y - 7z + 9 = 0
We also need to show that the points lie on the opposite side of the plane.
Normal of the plane is,
We know, the perpendicular distance of the position vector of a point
to the plane, p: ax + by + cz + d = 0 is given as:
Where
Thus, the perpendicular distance of the point to the plane 5x + 2y - 7z + 9 = 0 having normal is given by,
Hence, the perpendicular distance of the point to the plane 5x + 2y - 7z + 9 = 0 having normal
Therefore, |D1| = |D2|
However, D1 and D2 have different signs.
Therefore, the points A and B will lie on opposite sides of the plane.
Hence, we have successfully shown that the points are equidistant from the plane and lie on opposite sides of the plane.
Question:26
Answer:
Given,
And the position vectors
Therefore, the line passing through A and along AB will have the equation:
and the line passing through C and along CD will have equation
Now, PQ is a vector perpendicular to both AB and CD, such that Q lies on CD and P lies on AB. Thus, coordinates of P and Q will be of the form
Hence, the vector PQ will be given as
Now, since PQ is perpendicular to both, hence the dot products of AB.PQ and CD.PQ will be equal to 0.
AB. PQ = 0 and CD. PQ = 0
Solving (iii) and (iv), we get
Putting the value of in (i) we get,
Putting the value of in (ii) we get,
Hence, position vector of P and Q will be
Question:27
Answer:
We have given
, and
Thus, we get two cases:
=> -4m + 2m - n = 0 [from (i)]
=> n = 2m
, and
m = -2l
=> 2l + 2(-2l) - n = 0
=> 2l - 4l = n
=> n = -2l
Hence, the direction ratios of one line is proportional to -2m, m or -2m or direction ratios are (-2, 1, -2) and the direction ratios of another line is proportional to l, -2l, -2l, or direction ratios are (1, -2, -2)
Thus, the direction vectors of two lines are
Also, the angle between the two lines is given by:
Now,
Therefore, the lines have a 900 angle between them.
Question:28
Answer:
Let the direction vector of the 3 mutually perpendicular lines be
Let the direction vectors associated with direction cosines be
Since the lines associated with the direction vectors a, b, and c are mutually perpendicular, we get
(Since the dot product of two perpendicular vectors is 0)
=> …(1)
Similarly,
…(2)
Finally,
=> …(3)
Now, let us consider x, y and z as the angles made by direction vectors a, b, and c respectively with p.
Then,
We know, [since the sum of squares of direction cosines of a line = 1]
[from (1) and (2)]
Then, and
=> x = y = z = 0.
Therefore, the vector p makes equal angles with the vectors a, b and c.
Question:29
Distance of the point (α, β, y) is:
A. β B. |β| C. |β| + |y| D. √(α² + y²)
Answer:
Drawing a perpendicular from (α, β, y) to the y-axis gives us a foot of perpendicular with coordinates (0, β, 0).
Also, using the distance formula, we can calculate the distance between two points as:
Thus, the required distance
(Option D)
Question:30
If the direction cosines of a line are k, k, k, then:
A. k > 0
B. 0 < k < 1
C. k = 1
D. k = 1/√3 or -1/√3
Answer:
We know that the sum of squares of the direction cosines of a line = 1
=> k² + k² + k² = 1
=> 3k² = 1
(Option D)
Question:31
The distance of the plane from the origin is:
A. 1
B. 7
C. 1/7
D. None of these
Answer:
Given plane is
Let
=> n is a unit vector
Thus, the equation of the plane is of the form , where n is
the unit vector and d is the distance from the origin.
Comparing, we get d =1, hence the distance of the plane from origin is 1
(Option A)
Question:32
The sine of the angle between the straight line and
The plane 2x - 2y + z = 5 is:
A. 10/6√5
B. 4/5√2
C. 2√3/5
D. √2/10
Answer:
The equation of the line is given as
The direction vector of this line can be represented as
Also given is the equation of the plane 2x - 2y + z = 5
The normal to this plane is,
We also know that the angle between the line with the direction
vector b and the plane with the normal vector n is,
(Option D)
Question:33
The reflection of the point (α, β, γ) in the xy- plane is:
A. (α, β, 0)
B. (0, 0, γ)
C. (-α, -β, -γ)
D. (α, β, -γ)
Answer:
The equation of the XY plane is Z = 0
Given point (α, β, γ); if we draw a perpendicular from this point in the XY plane, the coordinates of the plane will be
(α, β, 0).
Let the reflection be (x, y, z)
So,
=> x = α
=> y = β
=> z = -γ
The reflection is: (α, β, -γ). (option D)
Question:34
The area of the quadrilateral ABCD, where A (0, 4, 1), B (2, 3, -1), C (4, 5, 0) and D (2, 6, 2) is equal to:
A. 9 square units
B. 18 square units
C. 27 square units
D. 81 square units
Answer:
Given, A (0, 4, 1), B (2, 3, -1), C (4, 5, 0) and D (2, 6, 2);
Since opposite vectors of this parallelogram are equal and opposite, ABCD is a parallelogram and we know the area of a parallelogram is | AB x CD|
= 9 square units (Option A).
Question:35
The locus represented by xy + yz = 0 is:
A. A pair of perpendicular lines
B. A pair of parallel lines
C. A pair of parallel planes
D. A pair of perpendicular planes
Answer:
Given, xy + yz = 0
=> x (y + z) = 0
=> x = 0 and y + z = 0
Clearly, the above equations are the equations of planes [of the form ax + by + cz + d = 0]
Also, x = 0 has the normal vector
And y + z = 0 has the normal vector
And the dot product of these two is
= 0
Hence, the planes are perpendicular (Option D).
Question:36
The plane 2x - 3y + 6z - 11 = 0 makes an angle with the x-axis. The value of α is:
A.
B.
C.
D.
Answer:
Given, the equation of the plane is 2x - 3y + 6z - 11 = 0.
The normal to this plane is,
Also, the x-axis has the direction vector
Also, we know that the angle between the line with direction vector b and the plane having the normal vector n is:
On comparing, we find (Option C)
Question:37
Answer:
We know, the equation of a plane cutting the coordinate axes at (a, 0, 0), (0, b, 0) and (0, 0, c) is given as
In this case, a = 2, b = 3, and c = 4.
Therefore, putting these values in the equation of the plane, we get
Question:38
Fill in the blanks: The direction cosines of the vector are _______.
Answer:
If l, m, and n are the direction cosines and the direction ratios of a line are a, b, and c, then we know:
According to the question,
a = 2, b = 2, c = -1
Then
Thus, the direction cosines are
l = 2/3, b = 2/3, c = -1/3
Question:39
Fill in the blanks: The vector equation of the line is __________.
Answer:
The equation of the line is given as
Clearly, the line passes through A (5, -4, 6) and has the direction ratios 3, 7, and 2.
Also, the position vector of A is
The direction vector of the given line will be:
Also, the vector equation of a line passing through the given point whose position vector is a and b is:
Hence, the required equation of the line will be:
Question:40
Fill in the blanks: The Cartesian equation of the plane is ______.
Answer:
By expanding the dot product given in the question, we can get the Cartesian equation of the plane.
Given:
Putting
Thus, the required Cartesian equation is x + y - z = 2.
Question:41
State True or False for the given statement:
The unit vector normal to the plane is
Answer:
Given, the equation of the plane is x + 2y + 3z - 6 = 0
The normal to this plane will be The unit vector of this normal is:
Therefore, the statement is True
Question:42
Answer:
The position vector of the first point (3, 4, -7) is
And the position vector of the second point(1, -1, 6):
Also, the vector equation of a line passing through two points with position vectors a and b is given by:
Thus, the required line equation is
Question:43
Answer:
To begin with, we convert the given plane equation to intercept form:
where a, b, and c are the intercepts on x, y, and z, axes respectively.
Given,
Dividing this equation on both sides by 4,
On comparison, we get the intercepts -2, 4/3, and -4/5 respectively.
Therefore, the statement is True.
Question:44
State True or False for the given statement: The angle between the line and the plane is
Answer:
We know, the angle between the plane with normal vector n and the line with direction vector b is denoted by:
Given equation of the line is
Hence, its direction vector will be:
Given equation of the plane is
Hence, its normal vector will be:
Thus, we have:
Therefore, the given statement is False.
Question:45
State True or False for the given statement:
The angle between the planes and is
Answer:
In vector form, if we take θ as the angle between the two planes
and
Then
Now, the given planes are and
Here, and
Therefore,
The statement is False.
Question:46
State True or False for the given statement:
The line lies in the plane
Answer:
The equation of the line is given as
Any point lying on this line will satisfy the plane equation if the line itself lies in the plane. Also, any point on this line will have a position vector:
Given equation of the plane
If we put a in the above equation,
Thus, the line does not lie in the given plane.
Therefore, the given statement is False.
Question:47
State True or False for the given statement.
The vector equation of the line is
Answer:
The given equation of the line is
It is clear from the equation that this line passes through A (5, -4, 6) and has the direction ratios 3, 7 and 2.
The position vector of A is
And the direction vector of the line will be
We know, the vector equation of a line that passes through a given point with position vector a and b is given as
Hence, the required line equation will be:
Thus, the statement is True.
Question:48
Answer:
We know, the equation of a line in Cartesian form is
, where a, b and c are the direction ratios and (x1, y1, z1) is a particular point on the line.
The given line is parallel to therefore it has 2, 1, 3 as direction ratios.
(a = 2, b = 1, c = 3)
The line passes through (5, -2, 4)
Substituting these values, we get the equation of line:
Thus, the given statement is False.
Question:49
Answer:
Let us take O as the origin, P as the foot of the perpendicular drawn from origin to the plane.
Then the position vector OP is:
The unit vector of n is:
Now, the equation of the plane with unit normal vector n and having a perpendicular drawn from the origin d is:
Therefore,
Equation of the given plane will be,
=> The given statement is True.
The main topics and sub topics covered in this chapter of are as follows:
Apply to Aakash iACST Scholarship Test 2024
· NCERT Exemplar Class 12 Maths chapter 11 solutions define Dimension as the standard measure of an object's size and shape.
· We would use the concept of Vector algebra to make three-dimensional geometry organised and straightforward.
· In Class 12 Maths NCERT exemplar solutions chapter 11, we will study the direction aspects of a line joining two points, including the direction cosines and the ratios, discuss the equations of lines and planes in space, measure the shortest distance between two lines and learn more about the Cartesian form analytical and geometric representation.
Chapter 1 | |
Chapter 2 | |
Chapter 3 | |
Chapter 4 | |
Chapter 5 | |
Chapter 6 | |
Chapter 7 | |
Chapter 8 | |
Chapter 9 | |
Chapter 10 | |
Chapter 11 | Three Dimensional Geometry |
Chapter 12 | |
Chapter 13 |
This entire chapter talks about the dimensional geometry which covered vector usage to measure and determine line, planes and angles.
Yes, for those who want a clear picture of how to solve questions in three-dimensional geometry, our NCERT exemplar Class 12 Maths solutions chapter 11 can be highly supportive.
The best way is to use these solutions as reference, while one is solving the questions for practicing.
Yes, these questions and NCERT exemplar Class 12 Maths solutions chapter 11 can be downloaded by using the webpage to PDF tool available online.
hello,
Yes you can appear for the compartment paper again since CBSE gives three chances to a candidate to clear his/her exams so you still have two more attempts. However, you can appear for your improvement paper for all subjects but you cannot appear for the ones in which you have failed.
I hope this was helpful!
Good Luck
Hello dear,
If you was not able to clear 1st compartment and now you giving second compartment so YES, you can go for your improvement exam next year but if a student receives an improvement, they are given the opportunity to retake the boards as a private candidate the following year, but there are some requirements. First, the student must pass all of their subjects; if they received a compartment in any subject, they must then pass the compartment exam before being eligible for the improvement.
As you can registered yourself as private candidate for giving your improvement exam of 12 standard CBSE(Central Board of Secondary Education).For that you have to wait for a whole year which is bit difficult for you.
Positive side of waiting for whole year is you have a whole year to preparing yourself for your examination. You have no distraction or something which may causes your failure in the exams. In whole year you have to stay focused on your 12 standard examination for doing well in it. By this you get a highest marks as a comparison of others.
Believe in Yourself! You can make anything happen
All the very best.
Hello Student,
I appreciate your Interest in education. See the improvement is not restricted to one subject or multiple subjects and we cannot say if improvement in one subject in one year leads to improvement in more subjects in coming year.
You just need to have a revision of all subjects what you have completed in the school. have a revision and practice of subjects and concepts helps you better.
All the best.
If you'll do hard work then by hard work of 6 months you can achieve your goal but you have to start studying for it dont waste your time its a very important year so please dont waste it otherwise you'll regret.
Yes, you can take admission in class 12th privately there are many colleges in which you can give 12th privately.
The field of biomedical engineering opens up a universe of expert chances. An Individual in the biomedical engineering career path work in the field of engineering as well as medicine, in order to find out solutions to common problems of the two fields. The biomedical engineering job opportunities are to collaborate with doctors and researchers to develop medical systems, equipment, or devices that can solve clinical problems. Here we will be discussing jobs after biomedical engineering, how to get a job in biomedical engineering, biomedical engineering scope, and salary.
Database professionals use software to store and organise data such as financial information, and customer shipping records. Individuals who opt for a career as data administrators ensure that data is available for users and secured from unauthorised sales. DB administrators may work in various types of industries. It may involve computer systems design, service firms, insurance companies, banks and hospitals.
A career as ethical hacker involves various challenges and provides lucrative opportunities in the digital era where every giant business and startup owns its cyberspace on the world wide web. Individuals in the ethical hacker career path try to find the vulnerabilities in the cyber system to get its authority. If he or she succeeds in it then he or she gets its illegal authority. Individuals in the ethical hacker career path then steal information or delete the file that could affect the business, functioning, or services of the organization.
The invention of the database has given fresh breath to the people involved in the data analytics career path. Analysis refers to splitting up a whole into its individual components for individual analysis. Data analysis is a method through which raw data are processed and transformed into information that would be beneficial for user strategic thinking.
Data are collected and examined to respond to questions, evaluate hypotheses or contradict theories. It is a tool for analyzing, transforming, modeling, and arranging data with useful knowledge, to assist in decision-making and methods, encompassing various strategies, and is used in different fields of business, research, and social science.
Individuals who opt for a career as geothermal engineers are the professionals involved in the processing of geothermal energy. The responsibilities of geothermal engineers may vary depending on the workplace location. Those who work in fields design facilities to process and distribute geothermal energy. They oversee the functioning of machinery used in the field.
Individuals who opt for a career as a remote sensing technician possess unique personalities. Remote sensing analysts seem to be rational human beings, they are strong, independent, persistent, sincere, realistic and resourceful. Some of them are analytical as well, which means they are intelligent, introspective and inquisitive.
Remote sensing scientists use remote sensing technology to support scientists in fields such as community planning, flight planning or the management of natural resources. Analysing data collected from aircraft, satellites or ground-based platforms using statistical analysis software, image analysis software or Geographic Information Systems (GIS) is a significant part of their work. Do you want to learn how to become remote sensing technician? There's no need to be concerned; we've devised a simple remote sensing technician career path for you. Scroll through the pages and read.
The role of geotechnical engineer starts with reviewing the projects needed to define the required material properties. The work responsibilities are followed by a site investigation of rock, soil, fault distribution and bedrock properties on and below an area of interest. The investigation is aimed to improve the ground engineering design and determine their engineering properties that include how they will interact with, on or in a proposed construction.
The role of geotechnical engineer in mining includes designing and determining the type of foundations, earthworks, and or pavement subgrades required for the intended man-made structures to be made. Geotechnical engineering jobs are involved in earthen and concrete dam construction projects, working under a range of normal and extreme loading conditions.
How fascinating it is to represent the whole world on just a piece of paper or a sphere. With the help of maps, we are able to represent the real world on a much smaller scale. Individuals who opt for a career as a cartographer are those who make maps. But, cartography is not just limited to maps, it is about a mixture of art, science, and technology. As a cartographer, not only you will create maps but use various geodetic surveys and remote sensing systems to measure, analyse, and create different maps for political, cultural or educational purposes.
Budget analysis, in a nutshell, entails thoroughly analyzing the details of a financial budget. The budget analysis aims to better understand and manage revenue. Budget analysts assist in the achievement of financial targets, the preservation of profitability, and the pursuit of long-term growth for a business. Budget analysts generally have a bachelor's degree in accounting, finance, economics, or a closely related field. Knowledge of Financial Management is of prime importance in this career.
The invention of the database has given fresh breath to the people involved in the data analytics career path. Analysis refers to splitting up a whole into its individual components for individual analysis. Data analysis is a method through which raw data are processed and transformed into information that would be beneficial for user strategic thinking.
Data are collected and examined to respond to questions, evaluate hypotheses or contradict theories. It is a tool for analyzing, transforming, modeling, and arranging data with useful knowledge, to assist in decision-making and methods, encompassing various strategies, and is used in different fields of business, research, and social science.
A Product Manager is a professional responsible for product planning and marketing. He or she manages the product throughout the Product Life Cycle, gathering and prioritising the product. A product manager job description includes defining the product vision and working closely with team members of other departments to deliver winning products.
An underwriter is a person who assesses and evaluates the risk of insurance in his or her field like mortgage, loan, health policy, investment, and so on and so forth. The underwriter career path does involve risks as analysing the risks means finding out if there is a way for the insurance underwriter jobs to recover the money from its clients. If the risk turns out to be too much for the company then in the future it is an underwriter who will be held accountable for it. Therefore, one must carry out his or her job with a lot of attention and diligence.
Individuals in the operations manager jobs are responsible for ensuring the efficiency of each department to acquire its optimal goal. They plan the use of resources and distribution of materials. The operations manager's job description includes managing budgets, negotiating contracts, and performing administrative tasks.
An investment director is a person who helps corporations and individuals manage their finances. They can help them develop a strategy to achieve their goals, including paying off debts and investing in the future. In addition, he or she can help individuals make informed decisions.
Welding Engineer Job Description: A Welding Engineer work involves managing welding projects and supervising welding teams. He or she is responsible for reviewing welding procedures, processes and documentation. A career as Welding Engineer involves conducting failure analyses and causes on welding issues.
A career as Transportation Planner requires technical application of science and technology in engineering, particularly the concepts, equipment and technologies involved in the production of products and services. In fields like land use, infrastructure review, ecological standards and street design, he or she considers issues of health, environment and performance. A Transportation Planner assigns resources for implementing and designing programmes. He or she is responsible for assessing needs, preparing plans and forecasts and compliance with regulations.
An expert in plumbing is aware of building regulations and safety standards and works to make sure these standards are upheld. Testing pipes for leakage using air pressure and other gauges, and also the ability to construct new pipe systems by cutting, fitting, measuring and threading pipes are some of the other more involved aspects of plumbing. Individuals in the plumber career path are self-employed or work for a small business employing less than ten people, though some might find working for larger entities or the government more desirable.
Individuals who opt for a career as construction managers have a senior-level management role offered in construction firms. Responsibilities in the construction management career path are assigning tasks to workers, inspecting their work, and coordinating with other professionals including architects, subcontractors, and building services engineers.
Urban Planning careers revolve around the idea of developing a plan to use the land optimally, without affecting the environment. Urban planning jobs are offered to those candidates who are skilled in making the right use of land to distribute the growing population, to create various communities.
Urban planning careers come with the opportunity to make changes to the existing cities and towns. They identify various community needs and make short and long-term plans accordingly.
Highway Engineer Job Description: A Highway Engineer is a civil engineer who specialises in planning and building thousands of miles of roads that support connectivity and allow transportation across the country. He or she ensures that traffic management schemes are effectively planned concerning economic sustainability and successful implementation.
Individuals who opt for a career as an environmental engineer are construction professionals who utilise the skills and knowledge of biology, soil science, chemistry and the concept of engineering to design and develop projects that serve as solutions to various environmental problems.
A Naval Architect is a professional who designs, produces and repairs safe and sea-worthy surfaces or underwater structures. A Naval Architect stays involved in creating and designing ships, ferries, submarines and yachts with implementation of various principles such as gravity, ideal hull form, buoyancy and stability.
Orthotists and Prosthetists are professionals who provide aid to patients with disabilities. They fix them to artificial limbs (prosthetics) and help them to regain stability. There are times when people lose their limbs in an accident. In some other occasions, they are born without a limb or orthopaedic impairment. Orthotists and prosthetists play a crucial role in their lives with fixing them to assistive devices and provide mobility.
A career in pathology in India is filled with several responsibilities as it is a medical branch and affects human lives. The demand for pathologists has been increasing over the past few years as people are getting more aware of different diseases. Not only that, but an increase in population and lifestyle changes have also contributed to the increase in a pathologist’s demand. The pathology careers provide an extremely huge number of opportunities and if you want to be a part of the medical field you can consider being a pathologist. If you want to know more about a career in pathology in India then continue reading this article.
Gynaecology can be defined as the study of the female body. The job outlook for gynaecology is excellent since there is evergreen demand for one because of their responsibility of dealing with not only women’s health but also fertility and pregnancy issues. Although most women prefer to have a women obstetrician gynaecologist as their doctor, men also explore a career as a gynaecologist and there are ample amounts of male doctors in the field who are gynaecologists and aid women during delivery and childbirth.
An oncologist is a specialised doctor responsible for providing medical care to patients diagnosed with cancer. He or she uses several therapies to control the cancer and its effect on the human body such as chemotherapy, immunotherapy, radiation therapy and biopsy. An oncologist designs a treatment plan based on a pathology report after diagnosing the type of cancer and where it is spreading inside the body.
The audiologist career involves audiology professionals who are responsible to treat hearing loss and proactively preventing the relevant damage. Individuals who opt for a career as an audiologist use various testing strategies with the aim to determine if someone has a normal sensitivity to sounds or not. After the identification of hearing loss, a hearing doctor is required to determine which sections of the hearing are affected, to what extent they are affected, and where the wound causing the hearing loss is found. As soon as the hearing loss is identified, the patients are provided with recommendations for interventions and rehabilitation such as hearing aids, cochlear implants, and appropriate medical referrals. While audiology is a branch of science that studies and researches hearing, balance, and related disorders.
The hospital Administrator is in charge of organising and supervising the daily operations of medical services and facilities. This organising includes managing of organisation’s staff and its members in service, budgets, service reports, departmental reporting and taking reminders of patient care and services.
For an individual who opts for a career as an actor, the primary responsibility is to completely speak to the character he or she is playing and to persuade the crowd that the character is genuine by connecting with them and bringing them into the story. This applies to significant roles and littler parts, as all roles join to make an effective creation. Here in this article, we will discuss how to become an actor in India, actor exams, actor salary in India, and actor jobs.
Individuals who opt for a career as acrobats create and direct original routines for themselves, in addition to developing interpretations of existing routines. The work of circus acrobats can be seen in a variety of performance settings, including circus, reality shows, sports events like the Olympics, movies and commercials. Individuals who opt for a career as acrobats must be prepared to face rejections and intermittent periods of work. The creativity of acrobats may extend to other aspects of the performance. For example, acrobats in the circus may work with gym trainers, celebrities or collaborate with other professionals to enhance such performance elements as costume and or maybe at the teaching end of the career.
Career as a video game designer is filled with excitement as well as responsibilities. A video game designer is someone who is involved in the process of creating a game from day one. He or she is responsible for fulfilling duties like designing the character of the game, the several levels involved, plot, art and similar other elements. Individuals who opt for a career as a video game designer may also write the codes for the game using different programming languages.
Depending on the video game designer job description and experience they may also have to lead a team and do the early testing of the game in order to suggest changes and find loopholes.
Radio Jockey is an exciting, promising career and a great challenge for music lovers. If you are really interested in a career as radio jockey, then it is very important for an RJ to have an automatic, fun, and friendly personality. If you want to get a job done in this field, a strong command of the language and a good voice are always good things. Apart from this, in order to be a good radio jockey, you will also listen to good radio jockeys so that you can understand their style and later make your own by practicing.
A career as radio jockey has a lot to offer to deserving candidates. If you want to know more about a career as radio jockey, and how to become a radio jockey then continue reading the article.
The word “choreography" actually comes from Greek words that mean “dance writing." Individuals who opt for a career as a choreographer create and direct original dances, in addition to developing interpretations of existing dances. A Choreographer dances and utilises his or her creativity in other aspects of dance performance. For example, he or she may work with the music director to select music or collaborate with other famous choreographers to enhance such performance elements as lighting, costume and set design.
A multimedia specialist is a media professional who creates, audio, videos, graphic image files, computer animations for multimedia applications. He or she is responsible for planning, producing, and maintaining websites and applications.
A career as social media manager involves implementing the company’s or brand’s marketing plan across all social media channels. Social media managers help in building or improving a brand’s or a company’s website traffic, build brand awareness, create and implement marketing and brand strategy. Social media managers are key to important social communication as well.
In a career as a copywriter, one has to consult with the client and understand the brief well. A career as a copywriter has a lot to offer to deserving candidates. Several new mediums of advertising are opening therefore making it a lucrative career choice. Students can pursue various copywriter courses such as Journalism, Advertising, Marketing Management. Here, we have discussed how to become a freelance copywriter, copywriter career path, how to become a copywriter in India, and copywriting career outlook.
Careers in journalism are filled with excitement as well as responsibilities. One cannot afford to miss out on the details. As it is the small details that provide insights into a story. Depending on those insights a journalist goes about writing a news article. A journalism career can be stressful at times but if you are someone who is passionate about it then it is the right choice for you. If you want to know more about the media field and journalist career then continue reading this article.
For publishing books, newspapers, magazines and digital material, editorial and commercial strategies are set by publishers. Individuals in publishing career paths make choices about the markets their businesses will reach and the type of content that their audience will be served. Individuals in book publisher careers collaborate with editorial staff, designers, authors, and freelance contributors who develop and manage the creation of content.
In a career as a vlogger, one generally works for himself or herself. However, once an individual has gained viewership there are several brands and companies that approach them for paid collaboration. It is one of those fields where an individual can earn well while following his or her passion.
Ever since internet costs got reduced the viewership for these types of content has increased on a large scale. Therefore, a career as a vlogger has a lot to offer. If you want to know more about the Vlogger eligibility, roles and responsibilities then continue reading the article.
Individuals in the editor career path is an unsung hero of the news industry who polishes the language of the news stories provided by stringers, reporters, copywriters and content writers and also news agencies. Individuals who opt for a career as an editor make it more persuasive, concise and clear for readers. In this article, we will discuss the details of the editor's career path such as how to become an editor in India, editor salary in India and editor skills and qualities.
Linguistic meaning is related to language or Linguistics which is the study of languages. A career as a linguistic meaning, a profession that is based on the scientific study of language, and it's a very broad field with many specialities. Famous linguists work in academia, researching and teaching different areas of language, such as phonetics (sounds), syntax (word order) and semantics (meaning).
Other researchers focus on specialities like computational linguistics, which seeks to better match human and computer language capacities, or applied linguistics, which is concerned with improving language education. Still, others work as language experts for the government, advertising companies, dictionary publishers and various other private enterprises. Some might work from home as freelance linguists. Philologist, phonologist, and dialectician are some of Linguist synonym. Linguists can study French, German, Italian.
The career of a travel journalist is full of passion, excitement and responsibility. Journalism as a career could be challenging at times, but if you're someone who has been genuinely enthusiastic about all this, then it is the best decision for you. Travel journalism jobs are all about insightful, artfully written, informative narratives designed to cover the travel industry. Travel Journalist is someone who explores, gathers and presents information as a news article.
Welding Engineer Job Description: A Welding Engineer work involves managing welding projects and supervising welding teams. He or she is responsible for reviewing welding procedures, processes and documentation. A career as Welding Engineer involves conducting failure analyses and causes on welding issues.
A quality controller plays a crucial role in an organisation. He or she is responsible for performing quality checks on manufactured products. He or she identifies the defects in a product and rejects the product.
A quality controller records detailed information about products with defects and sends it to the supervisor or plant manager to take necessary actions to improve the production process.
A Product Manager is a professional responsible for product planning and marketing. He or she manages the product throughout the Product Life Cycle, gathering and prioritising the product. A product manager job description includes defining the product vision and working closely with team members of other departments to deliver winning products.
A QA Lead is in charge of the QA Team. The role of QA Lead comes with the responsibility of assessing services and products in order to determine that he or she meets the quality standards. He or she develops, implements and manages test plans.
A metallurgical engineer is a professional who studies and produces materials that bring power to our world. He or she extracts metals from ores and rocks and transforms them into alloys, high-purity metals and other materials used in developing infrastructure, transportation and healthcare equipment.
An Azure Administrator is a professional responsible for implementing, monitoring, and maintaining Azure Solutions. He or she manages cloud infrastructure service instances and various cloud servers as well as sets up public and private cloud systems.
An AWS Solution Architect is someone who specializes in developing and implementing cloud computing systems. He or she has a good understanding of the various aspects of cloud computing and can confidently deploy and manage their systems. He or she troubleshoots the issues and evaluates the risk from the third party.
Careers in computer programming primarily refer to the systematic act of writing code and moreover include wider computer science areas. The word 'programmer' or 'coder' has entered into practice with the growing number of newly self-taught tech enthusiasts. Computer programming careers involve the use of designs created by software developers and engineers and transforming them into commands that can be implemented by computers. These commands result in regular usage of social media sites, word-processing applications and browsers.
A Product Manager is a professional responsible for product planning and marketing. He or she manages the product throughout the Product Life Cycle, gathering and prioritising the product. A product manager job description includes defining the product vision and working closely with team members of other departments to deliver winning products.
Individuals in the information security manager career path involves in overseeing and controlling all aspects of computer security. The IT security manager job description includes planning and carrying out security measures to protect the business data and information from corruption, theft, unauthorised access, and deliberate attack
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