NCERT Solutions for Class 12 Maths Exercise 10.4 Chapter 10 Vector Algebra – FREE PDF Download
Free PDF download of NCERT Solutions for Class 12 Maths Chapter 10 Exercise 10.4 (Ex 10.4) and all chapter exercises at one place prepared by expert teacher as per NCERT (CBSE) books guidelines. Class 12 Maths Chapter 10 Vector Algebra Exercise 10.4 Questions with Solutions to help you to revise complete Syllabus and Score More marks.
NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra (Ex 10.4) Exercise 10.4
1. Find 
if 
and 
and 
Expanding along first row,
= 
= 
2. Find a unit vector perpendicular to each of the vectors 
and where and
and 
where 
and 
On Adding 
= 
+ 
= 
On Subtracting 
= 
= 
Therefore, 
Expanding along first row = 
= 
Therefore, a unit vector perpendicular to both 
and 
is
= 
= 
= ±23iˆ∓23jˆ∓23kˆ±23i^∓23j^∓23k^
3. If a unit vector makes an angle with with and an acute angle with then find and hence, the components of .
and an acute angle 
then find
be a unit vector. ……….(i) 
Squaring both sides, 
……….(ii)
Given: Angle between vectors 
and iˆi^ is 
……….(iii)
Again, given Angel between vectors and jˆj^ is 
……….(iv)
Again, given Angel between vectors and kˆk^ is 
where 
is acute angle.
……….(v)
Putting the values of 
and 
in eq. (ii),
Since is acute angle, therefore 
is positive and hence 
From eq. (v), 
Putting values of and in eq. (i), 
Components of are coefficients of 
in
and angle
4. Show that (a→−b→)×(a→+b→)=2(a→×b→)(a→−b→)×(a→+b→)=2(a→×b→)
= 
= 
= 2(a→×b→)2(a→×b→) = R.H.S.
5. Find and if
Expanding along first row,
= 
Comparing the coefficients of on both sides, we have
……….(i)
……….(ii)
And 
……….(iii)
From eq. (ii), 
From eq. (iii), 
Putting the values of 
and 
in eq. (i),
0 = 0
Therefore, and λ=3.λ=3.
6. Given that and What can you conclude about the vectors and
What can you conclude about the vectors 
or 
or 
or 
or vector is perpendicular to 
…..(i)
Again, given 
or or 
or or vector and are collinear or parallel. …..(ii)
Since, vectors & are perpendicular to each other as well as parallel are not possible. ..(iii)
Therefore, from eq. (i), (ii) and (iii), either 
or 
and
7. Let the vectors be given as then show that
be given as 
then show that 
and 
Now L.H.S. = 
= 
+ 
[By Property of Determinants]
= 
= R.H.S.
8. If either and then Is the converse true? Justify your answer with an example.
then 
Is the converse true? Justify your answer with an example.
or 
or 
……….(i)
[Using eq. (i)]
[By definition of zero vector]
But the converse is not true.
Let 
is a non-zero vector.
Let 
is a non-zero vector.
But 
Taking 2 common from R3 = 
[
R2 and R3 are identical]
9. Find the area of the triangle with vertices A (1, 1, 2), B (2, 3, 5) and C (1, 5, 5).
are A (1, 1, 2), B (2, 3, 5) and C (1, 5, 5). Position vector of point A = (1, 1, 2) = 
Position vector of point B = (2, 3, 5) = 
Position vector of point C = (1, 5, 5) = 
Now 
= Position vector of point B – Position vector of point A
= 
= 
= 
And 
= Position vector of point C – Position vector of point A
=
=
= 
x = 
= 
= −6i→−3j→+4k→−6i→−3j→+4k→
Now Area of triangle ABC = 
= 
sq. units
10. Find the area of the parallelogram whose adjacent sides are determined by the vectors and
and 
and 
= 
= 
Now Area of parallelogram = 
= 
sq. units
11. Let the vectors and such that then is a unit vector, if the angle between and is:
then
(A) 
(B) 
(C) 
(D) 
and 
is a unit vector. 
, where is the angle between and
Therefore, option (B) is correct.
12. Area of a rectangle having vertices A, B, C and D with position vectors and respectively is:
and 
respectively is:
(A) 
(B) 1
(C) 2
(D) 4
Now
= Position vector of point B – Position vector of point A=
=
=
AB = 
And
= Position vector of point D – Position vector of point A=
=
=
AD = 
Area of rectangle ABCD = Length x Breadth = AB x AD = 2 x 1 = 2 sq. unitsTherefore, option (C) is correct.