## NCERT Solutions for Class 12 Maths Chapter 10 – Vector Algebra Ex 10.4

Find, if and.

We have,
and

#### Question 2:

Find a unit vector perpendicular to each of the vector and, where and.

#### Answer:

We have,
and
Hence, the unit vector perpendicular to each of the vectors and is given by the relation,

#### Question 3:

If a unit vector  makes an angleswith with and an acute angle θ with, then find θ and hence, the compounds of.

#### Answer:

Let unit vector  have (a1a2a3) components.
⇒
Since  is a unit vector, .
Also, it is given that  makes angleswith with , and an acute angle θ with
Then, we have:
Hence, and the components of  are.

Show that

#### Question 5:

Find λ and μ if .

#### Answer:

On comparing the corresponding components, we have:
Hence,

#### Question 6:

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

#### Answer:

Then,
(i) Either  or, or
(ii) Either  or, or
But,  and  cannot be perpendicular and parallel simultaneously.
Hence,  or.

#### Question 7:

Let the vectors given as . Then show that

#### Answer:

We have,
On adding (2) and (3), we get:
Now, from (1) and (4), we have:
Hence, the given result is proved.

#### Question 8:

If either  or, then. Is the converse true? Justify your answer with an example.

#### Answer:

Take any parallel non-zero vectors so that.
It can now be observed that:
Hence, the converse of the given statement need not be 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:

The vertices of triangle ABC are given as A (1, 1, 2), B (2, 3, 5), and
C (1, 5, 5).
The adjacent sidesand of ΔABC are given as:
Area of ΔABC
Hence, the area of ΔABC

#### Question 10:

Find the area of the parallelogram whose adjacent sides are determined by the vector .

#### Answer:

The area of the parallelogram whose adjacent sides are is.
Adjacent sides are given as:
Hence, the area of the given parallelogram is.

#### Question 11:

Let the vectors and be such that and, then is a unit vector, if the angle between and is
(A)  (B)  (C)  (D)

#### Answer:

It is given that.
We know that, where  is a unit vector perpendicular to both and and θ is the angle between and.
Now,  is a unit vector if.
Hence,  is a unit vector if the angle between and is.
The correct answer is B.

#### Question 12:

Area of a rectangle having vertices A, B, C, and D with position vectors and  respectively is
(A)  (B) 1
(C) 2 (D)

#### Answer:

The position vectors of vertices A, B, C, and D of rectangle ABCD are given as:
The adjacent sides and  of the given rectangle are given as:
⇒AB→×BC→=2Now, it is known that the area of a parallelogram whose adjacent sides are is.
Hence, the area of the given rectangle is
The correct answer is C.

Courtesy : CBSE