NCERT Solutions for Miscellaneous Exercise Chapter 10 Class 12 - Vector Algebra

Vector Algebra Class 12 Chapter 10 -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.