NCERT Solutions for Class 12 Maths Exercise 10.3 Chapter 10 Vector Algebra – FREE PDF Download
Free PDF download of NCERT Solutions for Class 12 Maths Chapter 10 Exercise 10.3 (Ex 10.3) 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 10.3 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 10.3) Exercise 10.3
1. Find the angle between two vectors 
and 
with magnitude 
and 2 respectively having 
and 
Let 
be the angle between the vector and 
We know that 
= 
= 
2. Find the angle between the vectors 
and
and 
and 
and 
Also 
= Product of coefficients of 
+ Product of coefficients of 
+ Product of coefficients 
= 
Let be the angle between the vector and
We know that
= 
= 
3. Find the projection of the vector on the vector
on the vector 
and 
Projection of vector and = 
= 
= 
If projection of vector and is zero, then vector is perpendicular to
4. Find the projection of the vector iˆ+3jˆ+7kˆi^+3j^+7k^ on the vector
and 
Projection of vector and = = 
= 
5. Show that each of the given three vectors is a unit vector:
Also show that they are mutually perpendicular to each other.
……….(i)
……….(ii)
……….(iii)
Each of the three given vectors 
is a unit vector.
From eq. (i) and (ii),
= 
and are perpendicular to each other.
From eq. (ii) and eq. (iii),
= 
and are perpendicular to each other.
From eq. (i) and (iii),
= 
and are perpendicular to each other.
Hence, are mutually perpendicular vectors.
6. Find and if and
and 
if 
and 
and 
……….(i) 
……….(ii)
Putting in eq. (ii),
Putting 
in eq (i),
7. Evaluate the product
= 
= 
= 
= 
8. Find the magnitude of two vectors and having the same magnitude such that the angle between them is and their scalar product is
having the same magnitude such that the angle between them is 
, angle (say) between and is 
and their scalar (i.e., dot) product = 
Putting and 
we have 
=
and 
9. Find if for a unit vector
if for a unit vector 
is a unit vector ……….(i)(x→−a→)(x→+a→)=12(x→−a→)(x→+a→)=12 
Putting from eq. (i), 
∣∣x→∣∣=13−−√|x→|=13
10. If and are such that is perpendicular to then find the value of
and 
is perpendicular to 
, 
and 
Now 
=
Again, = 
Since, 
is perpendicular to 
therefore, .
11. Show that is perpendicular to for any two non-zero vectors and
is perpendicular to 
for any two non-zero vectors
, where 
and 
Let 
Now 
= 
= 
= 
Putting, and ,
= 
= 0
Therefore, vectors 
and 
are perpendicular ot each other.
12. If and , then what can be concluded about the vector
Again 
0 = 0 for all (any vector )
Therefore, can be any vector.
13. If are unit vectors such that find the value of
find the value of 
are unit vectors.Therefore, 
and 
……….(i)Also given 
Putting the values from eq. (i), we get
14. If either vector or then . But the converse need not be true. Justify your answer with an example.
then
. Therefore by definition of zero vector, ……….(i) 
= 
[From eq. (i)]
Case II: Vector 
. Therefore by definition of zero vector, 
……….(ii)
= 
[From eq. (ii)]
But the converse is not true.
Justification: Let 
Therefore, 
Therefore, 
Again let 
Therefore, 
But 
Hence, here , but and .
15. If the vertices A, B, C of a triangle ABC are and (0, 1, 2) respectively, then find
and (0, 1, 2) respectively, then find 
and C (0, 1, 2) respectively. Position vector of point A = 
Position vector of point B = 
Position vector of point C = 
Now 
= Position vector of point A – Position vector of point B
= 
= 
……….(i)
And 
= Position vector of point C – Position vector of point B
= 
= 
……….(ii)
Let be the angle between the vectors and .
= 
[Using eq. (i) and (ii)]
16. Show that the points A (1, 2, 7), B (2, 6, 3) and C are collinear.
respectively. Position vector of point A = 
Position vector of point B = 
Position vector of point C = 
Now 
= Position vector of point B – Position vector of point A
= 
= 
……….(i)
And 
= Position vector of point C – Position vector of point A
= 
= 
……….(ii)
= 2. [Using eq. (i)]
Vectors and are collinear and parallel. 
Thus, points A, B and C are collinear.
And also vectors and have a common point A and hence can’t be parallel.
17. Show that the vectors and form the vertices of a right angled triangle.
and 
form the vertices of a right angled triangle.
Position vector of point A is 
Position vector of point B is 
and Position vector of point C is 
= Position vector of B – Position vector of A
= 
= 
= 
……….(i)
= Position vector of C – Position vector of B
= 
= 
= 
……….(ii)
= Position vector of C – Position vector of A
= 
= 
= ……….(iii)
Adding eq. (i) and (ii),
+ = + = = [Using eq. (iii)]
Therefore, by Triangle law of addition of vectors, points A, B, C are the vertices of a triangle ABC.
Now from eq. (i) and (ii),
. = 
= 
Again from eq. (ii) and (iii),
. = 
= 2 + 3 – 5 = 0
is perpendicular to .
Angle C is 
Therefore 
is a right angled at C.
Thus, A, B, C are the vertices of a right angled triangle.
18. If is a non-zero vector of magnitude and is a non-zero scalar, then is a unit vector if:
and 
is a non-zero scalar, then (A) 
(B) 
(C) 
(D) 
is a non-zero vector of magnitude 
Also given 
and 
is a unit vector.
Therefore, option (D) is correct.