Updated By SaraNextGen

On March 11, 2024, 11:35 AM

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 angles with  with  and an acute angle θ with , then find θ and hence, the compounds of .

Answer:

Let unit vector   have (a₁, a₂, a₃) components.

⇒ 

Since   is a unit vector,  .

Also, it is given that   makes angles with  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 sides and  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.