NCERT Solutions for Class 12 Maths Chapter 10 – Vector Algebra Ex 10.4
Page No 454:
Question 1:
Find
, if 
and
.
, if and.Answer:
We have,
and
Question 2:
Find a unit vector perpendicular to each of the vector 
and
, where 
and
.
and, where and.Answer:
We have,
and
Hence, the unit vector perpendicular to each of the vectors and is given by the relation,
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
.
makes an angleswith with and an acute angle θ with, then find θ and hence, the compounds of.Answer:
Let unit vector have (a1, a2, a3) components.
have (a1, a2, a3) components.
⇒ 
Since is a unit vector, 
.
is a unit vector, .
Also, it is given that makes angleswith with , and an acute angle θ with
makes angleswith with , and an acute angle θ with
Then, we have:
Hence,
and the components of are
.
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
?
and. What can you conclude about the vectors?Answer:
Then,
(i) Either 
or
, or 
or, or 
(ii) Either or, or 
or, or
But, 
and 
cannot be perpendicular and parallel simultaneously.
and cannot be perpendicular and parallel simultaneously.
Hence, or.
or.Question 7:
Let the vectors 
given as
. Then show that 
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.
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:
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
.
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
and be such that and, then is a unit vector, if the angle between and is
(A) 
(B) 
(C) 
(D) 
(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.
, where is a unit vector perpendicular to both and and θ is the angle between and.
Now, is a unit vector if
.
is a unit vector if.
Hence, is a unit vector if the angle between and is.
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
and respectively is
(A) 
(B) 1
(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)
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
.
is.
Hence, the area of the given rectangle is
The correct answer is C.
