NCERT Solutions for Class 12 Maths Chapter 10 – Vector Algebra Ex 10.4
Page No 454:
Question 1:
Find
, if
and
.



Answer:
We have,



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




Answer:
We have,



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

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: .png)


.png)
⇒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.