NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra

Class 12 Maths Chapter 10 NCERT Solutions Vector Algebra - Exercise: 10.1

Question:1 Represent graphically a displacement of 40 km, east of north.

Answer:

Represent graphically a displacement of 40 km, east of north.

N,S,E,W are all 4 direction north,south,east,west respectively.

is displacement vector which

= 40 km.

makes an angle of 30 degrees east of north as shown in the figure.

Question:2 (1) Classify the following measures as scalars and vectors.

10Kg

Answer:

10kg is a scalar quantity as it has only magnitude.

Question:2 (2) Classify the following measures as scalars and vectors. 2 meters north west

Answer:

This is a vector quantity as it has both magnitude and direction.

Question:2 (3) Classify the following measures as scalars and vectors.

Answer:

This is a scalar quantity as it has only magnitude.

Question:2 (4) Classify the following measures as scalars and vectors. 40 watt

Answer:

This is a scalar quantity as it has only magnitude.

Question:2 (5) Classify the following measures as scalars and vectors.

Answer:

This is a scalar quantity as it has only magnitude.

Question:2 (6) Classify the following measures as scalars and vectors.

Answer:

This is a Vector quantity as it has magnitude as well as direction.by looking at the unit, we conclude that measure is acceleration which is a vector.

Question:3 Classify the following as scalar and vector quantities.
(1) time period

Answer:

This is a scalar quantity as it has only magnitude.

Question:3 Classify the following as scalar and vector quantities.

(2) distance

Answer:

Distance is a scalar quantity as it has only magnitude.

Question:3 Classify the following as scalar and vector quantities.

(3) force

Answer:

Force is a vector quantity as it has both magnitude as well as direction.

Question:3 Classify the following as scalar and vector quantities.
(4) velocity

Answer:

Velocity is a vector quantity as it has both magnitude and direction.

Question:3 Classify the following as scalar and vector quantities.

(5) work done

Answer:

work done is a scalar quantity, as it is the product of two vectors.

Question:4 In Fig 10.6 (a square), identify the following vectors.
(1) Coinitial

Answer:

Since vector and vector are starting from the same point, they are coinitial.

Question:4 In Fig 10.6 (a square), identify the following vectors.
(2) Equal

Answer:

Since Vector and Vector both have the same magnitude and same direction, they are equal.

Question:4 In Fig 10.6 (a square), identify the following vectors.

(3) Collinear but not equal

Answer:

Since vector and vector have the same magnitude but different direction, they are colinear and not equal.

Question:5 Answer the following as true or false.
(1) and are collinear.

Answer:

True, and are collinear. they both are parallel to one line hence they are colinear.

Question:5 Answer the following as true or false.
(2) Two collinear vectors are always equal in magnitude.

Answer:

False, because colinear means they are parallel to the same line but their magnitude can be anything and hence this is a false statement.

Question:5 Answer the following as true or false.

(3) Two vectors having same magnitude are collinear.

Answer:

False, because any two non-colinear vectors can have the same magnitude.

Question:5 Answer the following as true or false.

(4) Two collinear vectors having the same magnitude are equal.

Answer:

False, because two colinear vectors with the same magnitude can have opposite direction

Class 12 Maths Chapter 10 NCERT Solutions Vector Algebra-Exercise: 10.2

Question:1 Compute the magnitude of the following vectors:

(1)

Answer:

Here

Magnitude of

Question:1 Compute the magnitude of the following vectors:

(2)

Answer:

Here,

Magnitude of

Question:1 Compute the magnitude of the following vectors:

(3)

Answer:

Here,

Magnitude of

Question:2 Write two different vectors having same magnitude

Answer:

Two different Vectors having the same magnitude are

The magnitude of both vector

Question:3 Write two different vectors having same direction.

Answer:

Two different vectors having the same direction are:

Question:4 Find the values of x and y so that the vectors and are equal.

Answer:

will be equal to when their corresponding components are equal.

Hence when,

and

Question:5 Find the scalar and vector components of the vector with initial point (2, 1) and terminal point (– 5, 7).

Answer:

Let point P = (2, 1) and Q = (– 5, 7).

Now,

Hence scalar components are (-7,6) and the vector is

Question:6 Find the sum of the vectors

Answer:

Given,

Now, The sum of the vectors:

Question:7 Find the unit vector in the direction of the vector

Answer:

Given

Magnitude of

A unit vector in the direction of

Question:8 Find the unit vector in the direction of vector , where P and Q are the points (1, 2, 3) and (4, 5, 6), respectively.

Answer:

Given P = (1, 2, 3) and Q = (4, 5, 6)

A vector in direction of PQ

Magnitude of PQ

Now, unit vector in direction of PQ

Question:9 For given vectors, and , find the unit vector in the direction of the vector .

Answer:

Given

Now,

Now a unit vector in the direction of

Question:10 Find a vector in the direction of vector which has magnitude 8 units.

Answer:

Given a vector

the unit vector in the direction of

A vector in direction of and whose magnitude is 8 =

Question:11 Show that the vectors and are collinear.

Answer:

Let

It can be seen that

Hence here

As we know

Whenever we have , the vector and will be colinear.

Here

Hence vectors and are collinear.

Question:12 Find the direction cosines of the vector

Answer:

Let

Hence direction cosine of are

Question:13 Find the direction cosines of the vector joining the points A(1, 2, –3) and B(–1, –2, 1), directed from A to B.

Answer:

Given

point A=(1, 2, –3)

point B=(–1, –2, 1)

Vector joining A and B Directed from A to B

Hence Direction cosines of vector AB are

Question:14 Show that the vector is equally inclined to the axes OX, OY and OZ.

Answer:

Let

Hence direction cosines of this vectors is

Let , and be the angle made by x-axis, y-axis and z- axis respectively

Now as we know,

,

Hence Given vector is equally inclined to axis OX,OY and OZ.

Question:15 (1) Find the position vector of a point R which divides the line joining two points P and Q whose position vectors are and respectively, in the ratio 2 : 1 internally

Answer:

As we know

The position vector of the point R which divides the line segment PQ in ratio m:n internally:

Here

position vector os P = =

the position vector of Q =

m:n = 2:1

And Hence

Question:15 (2) Find the position vector of a point R which divides the line joining two points P and Q whose position vectors are and respectively, in the ratio 2 : 1 externally

Answer:

As we know

The position vector of the point R which divides the line segment PQ in ratio m:n externally:

Here

position vector os P = =

the position vector of Q =

m:n = 2:1

And Hence

Question:16 Find the position vector of the mid point of the vector joining the points P(2, 3, 4) and Q(4, 1, –2).

Answer:

Given

The position vector of point P =

Position Vector of point Q =

The position vector of R which divides PQ in half is given by:

Question:17 Show that the points A, B and C with position vectors, , respectively form the vertices of a right angled triangle.

Answer:

Given

the position vector of A, B, and C are

Now,

AS we can see

Hence ABC is a right angle triangle.

Question:18 In triangle ABC (Fig 10.18), which of the following is not true:

Answer:

From triangles law of addition we have,

From here

also

Also

Hence options A,B and D are true SO,

Option C is False.

Question:19 If are two collinear vectors, then which of the following are incorrect:
(A) for some saclar
(B)
(C) the respective components of are not proportional
(D) both the vectors have same direction, but different magnitudes.

Answer:

If two vectors are collinear then, they have same direction or are parallel or anti-parallel.
Therefore,
They can be expressed in the form where a and b are vectors and is some scalar quantity.

Therefore, (a) is true.
Now,
(b) is a scalar quantity so its value may be equal to

Therefore,
(b) is also true.

C) The vectors and are proportional,
Therefore, (c) is not true.

D) The vectors and can have different magnitude as well as different directions.

Therefore, (d) is not true.

Therefore, the correct options are (C) and (D).

Class 12 Maths Chapter 10 NCERT Solutions Vector Algebra - Exercise: 10.3

Question:1 Find the angle between two vectors with magnitudes , respectively having .

Answer:

Given

As we know

where is the angle between two vectors

So,

Hence the angle between the vectors is .

Question:2 Find the angle between the vectors

Answer:

Given two vectors

Now As we know,

The angle between two vectors and is given by

Hence the angle between

Question:3 Find the projection of the vector on the vector

Answer:

Let

Projection of vector on

Hence, Projection of vector on is 0.

Question:4 Find the projection of the vector on the vector

Answer:

Let

The projection of on is

Hence, projection of vector on is

Question:5 Show that each of the given three vectors is a unit vector: Also, show that they are mutually perpendicular to each other.

Answer:

Given

Now magnitude of

Hence, they all are unit vectors.

Now,

Hence all three are mutually perpendicular to each other.

Question:6 Find , if .

Answer:

Given in the question

Since

So, answer of the question is

Question:7 Evaluate the product .

Answer:

To evaluate the product

Question:8 Find the magnitude of two vectors , having the same magnitude and such that the angle between them is and their scalar product is 1/2

Answer:

Given two vectors

Now Angle between

Now As we know that

Hence, the magnitude of two vectors

Question:9 Find , if for a unit vector

Answer:

Given in the question that

And we need to find

So the value of is

Question:10 If are such that is perpendicular to , then find the value of

Answer:

Given in the question is

and is perpendicular to

and we need to find the value of ,

so the value of -

As is perpendicular to

the value of ,

Question:11 Show that is perpendicular to , for any two nonzero vectors .

Answer:

Given in the question that -

are two non-zero vectors

According to the question

Hence is perpendicular to .

Question:12 If , then what can be concluded about the vector ?

Answer:

Given in the question

Therefore is a zero vector. Hence any vector will satisfy

Question:13 If are unit vectors such that , find the value of

Answer:

Given in the question

are unit vectors

and

and we need to find the value of

Answer- the value of is

Question:14 If either vector . But the converse need not be true. Justify your answer with an example

Answer:

Let

we see that

we now observe that

Hence here converse of the given statement is not true.

Question:15 If the vertices A, B, C of a triangle ABC are (1, 2, 3), (–1, 0, 0), (0, 1, 2), respectively, then find is the angle between the vectors .

Answer:

Given points,

A=(1, 2, 3),

B=(–1, 0, 0),

C=(0, 1, 2),

As need to find Angle between

Hence angle between them ;

Answer - Angle between the vectors is

Question:16 Show that the points A(1, 2, 7), B(2, 6, 3) and C(3, 10, –1) are collinear.

Answer:

Given in the question

A=(1, 2, 7), B=(2, 6, 3) and C(3, 10, –1)

To show that the points A(1, 2, 7), B(2, 6, 3) and C(3, 10, –1) are collinear

As we see that

Hence point A, B , and C are colinear.

Question:17 Show that the vectors form the vertices of a right angled triangle.

Answer:

Given the position vector of A, B , and C are

To show that the vectors form the vertices of a right angled triangle

Here we see that

Hence A,B, and C are the vertices of a right angle triangle.

Question:18 If is a nonzero vector of magnitude ‘a’ and a nonzero scalar, then is unit vector if

Answer:

Given is a nonzero vector of magnitude ‘a’ and a nonzero scalar

is a unit vector when

Hence the correct option is D.

Class 12 vector algebra NCERT solutions - Exercise: 10.4

Question:1 Find

Answer:

Given in the question,

and we need to find

Now,

So the value of is

Question:2 Find a unit vector perpendicular to each of the vector , where

Answer:

Given in the question

Now , A vector which perpendicular to both is

And a unit vector in this direction :

Hence Unit vector perpendicular to each of the vector is .

Question:3 If a unit vector makes angles with with and an acute angle with then find and hence, the components of .

Answer:

Given in the question,

angle between and :

angle between and

angle with and :

Now, As we know,

Now components of are:

Question:4 Show that

Answer:

To show that

LHS=

As product of a vector with itself is always Zero,

As cross product of a and b is equal to negative of cross product of b and a.

= RHS

LHS is equal to RHS, Hence Proved.

Question:5 Find and if

Answer:

Given in the question

and we need to find values of and

From Here we get,

From here, the value of and is

Question:6 Given that and . What can you conclude about the vectors ?

Answer:

Given in the question

and

When , either are perpendicular to each other

When either are parallel to each other

Since two vectors can never be both parallel and perpendicular at same time,we conclude that

Question:7 Let the vectors be given as Then show that

Answer:

Given in the question

We need to show that

Now,

Now

Hence they are equal.

Question:8 If either then . Is the converse true? Justify your answer with an example.

Answer:

No, the converse of the statement is not true, as there can be two non zero vectors, the cross product of whose are zero. they are colinear vectors.

Consider an example

Here

Hence converse of the given statement is not true.

Question:9 Find the area of the triangle with vertices A(1, 1, 2), B(2, 3, 5) and C(1, 5, 5).

Answer:

Given in the question

vertices A=(1, 1, 2), B=(2, 3, 5) and C=(1, 5, 5). and we need to find the area of the triangle

Now as we know

Area of triangle

The area of the triangle is square units

Question:10 Find the area of the parallelogram whose adjacent sides are determined by the vectors and .

Answer:

Given in the question

Area of parallelogram with adjescent side and ,

The area of the parallelogram whose adjacent sides are determined by the vectors and is

Question:11 Let the vectors be such that , then is a unit vector, if the angle between is

Answer:

Given in the question,

As given is a unit vector, which means,

Hence the angle between two vectors is . Correct option is B.

Question:12 Area of a rectangle having vertices A, B, C and D with position vectors

(A)1/2

(B) 1

(C) 2

(D) 4

Answer:

Given 4 vertices of rectangle are

Now,

Area of the Rectangle

Hence option C is correct.

Class 12 vector algebra NCERT Solutions - Miscellaneous Exercise

Question:1 Write down a unit vector in XY-plane, making an angle of with the positive direction of x-axis.

Answer:

As we know

a unit vector in XY-Plane making an angle with x-axis :

Hence for

Answer- the unit vector in XY-plane, making an angle of with the positive direction of x-axis is

Question:2 Find the scalar components and magnitude of the vector joining the points

Answer:

Given in the question

And we need to finrd the scalar components and magnitude of the vector joining the points P and Q

Magnitiude of vector PQ

Scalar components are

Question:3 A girl walks 4 km towards west, then she walks 3 km in a direction east of north and stops. Determine the girl’s displacement from her initial point of departure.

Answer:

As the girl walks 4km towards west

Position vector =

Now as she moves 3km in direction 30 degree east of north.

hence final position vector is;

Question:4 If , then is it true that ? Justify your answer.

Answer:

No, if then we can not conclude that .

the condition satisfies in the triangle.

also, in a triangle,

Since, the condition is contradicting with the triangle inequality, if then we can not conclude that

Question:5 Find the value of x for which is a unit vector.

Answer:

Given in the question,

a unit vector,

We need to find the value of x

The value of x is

Question:6 Find a vector of magnitude 5 units, and parallel to the resultant of the vectors

Answer:

Given two vectors

Resultant of and :

Now, a unit vector in the direction of

Now, a unit vector of magnitude in direction of

Hence the required vector is

Question:7 If , find a unit vector parallel to the vector .

Answer:

Given in the question,

Now,

let vector

Now, a unit vector in direction of

Now,

A unit vector parallel to

OR

Question:8 Show that the points A(1, – 2, – 8), B(5, 0, –2) and C(11, 3, 7) are collinear, and find the ratio in which B divides AC.

Answer:

Given in the question,

points A(1, – 2, – 8), B(5, 0, –2) and C(11, 3, 7)

now let's calculate the magnitude of the vectors

As we see that AB = BC + AC, we conclude that three points are colinear.

we can also see from here,

Point B divides AC in the ratio 2 : 3.

Question:9 Find the position vector of a point R which divides the line joining two points P and Q whose position vectors are externally in the ratio 1: 2. Also, show that P is the mid point of the line segment RQ.

Answer:

Given, two vectors

the point R which divides line segment PQ in ratio 1:2 is given by

Hence position vector of R is .

Now, Position vector of the midpoint of RQ

which is the position vector of Point P . Hence, P is the mid-point of RQ

Question:10 The two adjacent sides of a parallelogram are . Find the unit vector parallel to its diagonal. Also, find its area.

Answer:

Given, two adjacent sides of the parallelogram

The diagonal will be the resultant of these two vectors. so

resultant R:

Now unit vector in direction of R

Hence unit vector along the diagonal of the parallelogram

Now,

Area of parallelogram

Hence the area of the parallelogram is .

Question:11 Show that the direction cosines of a vector equally inclined to the axes OX, OY and OZ are

Answer:

Let a vector is equally inclined to axis OX, OY and OZ.

let direction cosines of this vector be

Now

Hence direction cosines are:

Question:12 Let . Find a vector which is perpendicular to both

Answer:

Given,

Let

now, since it is given that d is perpendicular to and , we got the condition,

and

And

And

here we got 2 equation and 3 variable. one more equation will come from the condition:

so now we have three equation and three variable,

On solving this three equation we get,

,

Hence Required vector :

Question:13 The scalar product of the vector with a unit vector along the sum of vectors and is equal to one. Find the value of .

Answer:

Let, the sum of vectors and be

unit vector along

Now, the scalar product of this with

Question:14 If are mutually perpendicular vectors of equal magnitudes, show that the vector is equally inclined to .

Answer:

Given

and

Now, let vector is inclined to at respectively.

Now,

Now, Since,

Hence vector is equally inclined to .

Question:15 Prove that , if and only if are perpendicular, given

Answer:

Given in the question,

are perpendicular and we need to prove that

LHS=

if are perpendicular,

= RHS

LHS ie equal to RHS

Hence proved.

Question:16 Choose the correct answer If is the angle between two vectors , then only when

Answer:

Given in the question

is the angle between two vectors

this will satisfy when

Hence option B is the correct answer.

Question:17 Choose the correct answer. Let be two unit vectors and is the angle between them. Then is a unit vector if

Answer:

Gicen in the question

be two unit vectors and is the angle between them

also

Then is a unit vector if

Hence option D is correct.

Question:18 The value of is

(A) 0

(B) –1

(C) 1

(D) 3

Answer:

To find the value of

Hence option C is correct.

Question:19 Choose the correct. If is the angle between any two vectors , then when
is equal to

Answer:

Given in the question

is the angle between any two vectors and

To find the value of

Hence option D is correct.