## EXERCISE 4.4

#### Page No 454:

#### Question 1:

Find, if and.

#### Answer:

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 (*a*_{1}, *a*_{2}, *a*_{3}) 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.

#### Question 4:

Show that

#### Answer:

#### 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

#### Page No 455:

#### 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.