The NCERT Exemplar textbooks are important for students in order to attain strong conceptual knowledge. They can now access the NCERT Exemplar Solutions, available subject-wise, for guidance to solve the exemplar problems. All the solutions are crafted to help students prepare well for the board exam. Further, all solutions are created by subject matter experts, following the latest CBSE syllabus.
The 10th Chapter of NCERT Exemplar Solutions for Class 12 Mathematics is Vector Algebra. Here, students will learn the introduction to vectors, their types, addition and multiplication performed on vectors, components of the vector, section formula, scalar product, projection of a vector on a line and vector product. To attain a strong grip over the concepts of this chapter, students can make use of the solutions PDF of NCERT Exemplar Solutions for Class 12 Maths Chapter 10 Vector Algebra from the link given below.
Download the PDF of NCERT Exemplar Solutions for Class 12 Maths Chapter 10 Vector Algebra
Access Answers to the NCERT Exemplar Class 12 Maths Chapter 10 Vector Algebra
Exercise 10.3 Page No: 215
1. Find the unit vector in the direction of sum of vectors
Solution:
Given vectors are,
2. If find the unit vector in the direction of
Solution:
3. Find a unit vector in the direction of where P and Q have co-ordinates (5, 0, 8) and (3, 3, 2), respectively.
Solution:
Given coordinates are P(5, 0, 8) and Q(3, 3, 2).
4. If are the position vectors of A and B, respectively, find the position vector of a point C in BA produced such that BC = 1.5 BA.
Solution:
5. Using vectors, find the value of k such that the points (k, – 10, 3), (1, –1, 3) and (3, 5, 3) are collinear.
Solution:
Let the given points be A(k, – 10, 3), B(1, –1, 3) and C(3, 5, 3).
6. A vector 
is inclined at equal angles to the three axes. If the magnitude of 
is 2√3 units, find
.
Solution:
As the vector 
makes equal angles with the axes, their direction cosines should also be same
So, l = m = n
And we know that,
l2 + m2 + n2 = 1 ⇒ l2 + l2 + l2 = 1
3l2 = 1
l = ± 1/√3
7. A vector 
has magnitude 14 and direction ratios 2, 3, – 6. Find the direction cosines and components of
, given that
makes an acute angle with x-axis.
8. Find a vector of magnitude 6, which is perpendicular to both the vectors 
and 
Solution:
9. Find the angle between the vectors
Solution:
10. If 
, show that 
. Interpret the result geometrically?
Solution:
11. Find the sine of the angle between the vectors 
and 
Solution:
Thus, sin θ = 2/√7
12. If A, B, C, D are the points with position vectors 
respectively, find the projection of 
along
Solution:
We have,
13. Using vectors, find the area of the triangle ABC with
vertices A(1, 2, 3), B(2, – 1, 4) and C(4, 5, – 1).
Solution:
Given vertices A(1, 2, 3), B(2, – 1, 4) and C(4, 5, – 1).
14. Using vectors, prove that the parallelogram on the same base and between the same parallels are equal in area.
Solution:
Let’s consider ABCD and ABFE be two parallelograms on the same base AB and between same parallel lines AB and DF.
– Hence proved.
Long Answer (L.A.)
15. Prove that in any triangle ABC, , where a, b, c are the
magnitudes of the sides opposite to the vertices A, B, C, respectively.
Solution:
In triangle ABC, the components of c are c cos A and c sin A.
16. If 
determine the vertices of a triangle, show that gives the
vector area of the triangle. Hence deduce the condition that the three points 
are collinear. Also find the unit vector normal to the plane of the triangle.
Solution:
17. Show that area of the parallelogram whose diagonals are given by 
and 
is . Also find the area of the parallelogram whose diagonals are 
and 
.
Solution:
Let’s take ABCD to be a parallelogram such that
18. If 
find a vector 
such that 
Solution:
Objective Type Questions
Choose the correct answer from the given four options in each of the Exercises from 19 to 33 (M.C.Q)
19. The vector in the direction of the vector 
that has magnitude 9 is
Solution:
The correct option is (C).
20. The position vector of the point which divides the join of points 
in the ratio 3 : 1 is
Solution:
The correct option is (D).
The given vectors are in the ratio 3: 1
21. The vector having initial and terminal points as (2, 5, 0) and (–3, 7, 4), respectively is 
Solution:
The correct option is (C).
Let A and B be two points whose coordinates are given as (2, 5, 0) and (-3, 7, 4)
So, we have
22. The angle between two vectors 
and 
with magnitudes √3 and 4, respectively, and 
is
Solution:
The correct option is (B).
Comments