MCQ Questions for Class 12 Maths Chapter 10 Vector Algebra with Answers

Do you need some help in preparing for your upcoming Maths exam? We’ve compiled a list of MCQ Questions for Class 12 Maths with Answers to get you started with the subject, Vector Algebra Class 12 MCQs Questions with Answers. You can download NCERT MCQ Questions for Class 12 Maths Chapter 10 Vector Algebra with Answers Pdf free download and learn how smart students improve problem-solving skills well ahead. So, ace up your preparation with Class 12 Maths Chapter 10 Vector Algebra Objective Questions.

Vector Algebra Class 12 MCQs Questions with Answers

Don’t forget to practice the multitude of MCQ Questions on Vector Algebra Class 12 with answers so you can apply your skills during the exam.

Question 1.
In ΔABC, which of the following is not true?

(a) $$\vec { AB}$$ + $$\vec { BC}$$ + $$\vec { CA}$$ = $$\vec { 0}$$
(b) $$\vec { AB}$$ + $$\vec { BC}$$ – $$\vec { AC}$$ = $$\vec { 0}$$
(c) $$\vec { AB}$$ + $$\vec { BC}$$ – $$\vec { CA}$$ = $$\vec { 0}$$
(d) $$\vec { AB}$$ – $$\vec { CB}$$ + $$\vec { CA}$$ = $$\vec { 0}$$

Answer: (c) $$\vec { AB}$$ + $$\vec { BC}$$ – $$\vec { CA}$$ = $$\vec { 0}$$

Question 2.
If $$\vec a$$ and $$\vec b$$ are two collinear vectors, then which of the following are incorrect:
(a) $$\vec b$$ = λ$$\vec a$$ tor some scalar λ.
(b) $$\vec a$$ = ±$$\vec b$$
(c) the respective components of $$\vec a$$ and $$\vec b$$ are proportional
(d) both the vectors $$\vec a$$ and $$\vec b$$ have the same direction, but different magnitudes.

Answer: (d) both the vectors $$\vec a$$ and $$\vec b$$ have the same direction, but different magnitudes.

Question 3.
If a is a non-zero vector of magnitude ‘a’ and λa non-zero scalar, then λ$$\vec a$$ is unit vector if:
(a) λ = 1
(b) λ = -1
(c) a = |λ|
(d) a = $$\frac { 1 }{|λ|}$$

Answer: (d) a = $$\frac { 1 }{|λ|}$$

Question 4.
Let λ be any non-zero scalar. Then for what possible values of x, y and z given below, the vectors 2$$\hat i$$ – 3$$\hat j$$ + 4$$\hat k$$ and x$$\hat i$$ – y$$\hat j$$ + z$$\hat k$$ are perpendicular:
(a) x = 2λ. y = λ, z = λ
(b) x = λ, y = 2λ, z = -λ
(c) x = -λ, y = 2λ, z = λ
(d) x = -λ, y = -2λ, z = λ.

Answer: (c) x = -λ, y = 2λ, z = λ

Question 5.
Let the vectors $$\vec a$$ and $$\vec b$$ be such that |$$\vec a$$| = 3 and |$$\vec b$$| = $$\frac { √2 }{3}$$, then $$\vec a$$ × $$\vec b$$ is a unit vector if the angle between $$\vec a$$ and $$\vec b$$ is:
(a) $$\frac { π }{6}$$
(b) $$\frac { π }{4}$$
(c) $$\frac { π }{3}$$
(d) $$\frac { π }{2}$$

Answer: (b) $$\frac { π }{4}$$

Question 6.
Area of a rectangle having vertices
A(-$$\hat i$$ + $$\frac { 1 }{2}$$ $$\hat j$$ + 4$$\hat k$$),
B($$\hat i$$ + $$\frac { 1 }{2}$$ $$\hat j$$ + 4$$\hat k$$),
C($$\hat i$$ – $$\frac { 1 }{2}$$ $$\hat j$$ + 4$$\hat k$$),
D(-$$\hat i$$ – $$\frac { 1 }{2}$$ $$\hat j$$ + 4$$\hat k$$) is
(a) $$\frac { 1 }{2}$$ square unit
(b) 1 square unit
(c) 2 square units
(d) 4 square units.

Question 7.
If θ is the angle between two vectors $$\vec a$$, $$\vec b$$, then $$\vec a$$.$$\vec b$$ ≥ 0 only when
(a) 0 < θ < $$\frac { π }{2}$$
(b) 0 ≤ θ ≤ $$\frac { π }{2}$$
(c) 0 < θ < π
(d) 0 ≤ θ ≤ π

Answer: (b) 0 ≤ θ ≤ $$\frac { π }{2}$$

Question 8.
Let $$\vec a$$ and $$\vec b$$ be two unit vectors and 6 is the angle between them. Then $$\vec a$$ + $$\vec b$$ is a unit vector if:
(a) θ = $$\frac { π }{4}$$
(b) θ = $$\frac { π }{3}$$
(c) θ = $$\frac { π }{2}$$
(d) θ = $$\frac { 2π }{3}$$

Answer: (d) θ = $$\frac { 2π }{3}$$

Question 9.
If {$$\hat i$$, $$\hat j$$, $$\hat k$$} are the usual three perpendicular unit vectors, then the value of:
$$\hat i$$.($$\hat j$$ × $$\hat k$$) + $$\hat j$$.($$\hat i$$ × $$\hat k$$) + $$\hat k$$.($$\hat i$$ × $$\hat j$$) is
(a) 0
(b) -1
(c) 1
(d) 3

Question 10.
If θ is the angle between two vectors $$\vec a$$ and $$\vec b$$, then |$$\vec a$$.$$\vec b$$| = |$$\vec a$$ × $$\vec b$$| when θ is equal to:
(a) 0
(b) $$\frac { π }{4}$$
(c) $$\frac { π }{2}$$
(d) π

Answer: (b) $$\frac { π }{4}$$

Question 11.
The area of the triangle whose adjacent sides are
$$\vec a$$ = 3$$\hat i$$ + $$\hat j$$ + 4$$\hat k$$ and $$\vec b$$ = $$\hat i$$ – $$\hat j$$ + $$\hat k$$ is
(a) $$\frac { 1 }{2}$$ $$\sqrt{ 42 }$$
(b) 42
(c) $$\sqrt{ 42 }$$
(d) $$\sqrt{ 21 }$$

Answer: (a) $$\frac { 1 }{2}$$ $$\sqrt{ 42 }$$

Question 12.
The magnitude of the vector 6$$\hat i$$ + 2$$\hat j$$ + 3$$\hat k$$ is
(a) 5
(b) 7
(c) 12
(d) 1.

Question 13.
The vector with initial point P (2, -3, 5) and terminal point Q (3, -4, 7) is
(a) $$\hat i$$ – $$\hat j$$ + 2$$\hat k$$
(b) 5$$\hat i$$ – 7$$\hat j$$ + 12$$\hat k$$
(c) –$$\hat i$$ + $$\hat j$$ – 2$$\hat k$$
(d) None of these.

Answer: (a) $$\hat i$$ – $$\hat j$$ + 2$$\hat k$$

Question 14.
The angle between the vectors $$\hat i$$ – $$\hat j$$ and $$\hat j$$ – $$\hat k$$ is
(a) $$\frac { π }{3}$$
(b) $$\frac { 2π }{3}$$
(c) –$$\frac { π }{3}$$
(d) $$\frac { 5π }{6}$$

Answer: (b) $$\frac { 2π }{3}$$

Question 15.
The value of ‘λ’ for which the two vectors:
2$$\hat i$$ – $$\hat j$$ + 2$$\hat k$$ and 3$$\hat i$$ + λ$$\hat j$$ + $$\hat k$$ are perpendicular is
(a) 2
(b) 4
(c) 6
(d) 8.

Question 16.
If |$$\vec a$$| = 8, |$$\vec b$$| = 3 and |$$\vec a$$ × $$\vec b$$|= 12, then value of $$\vec a$$.$$\vec b$$ is
(a) 6√3
(b) 8√3
(c) 12√3
(d) None of these.

Question 17.
The non-zero vectors $$\vec a$$, $$\vec b$$ and $$\vec c$$ are related by $$\vec a$$ = 8$$\vec b$$ and $$\vec c$$ = -7$$\vec b$$. Then the angle between $$\vec a$$ and $$\vec c$$ is
(a) π
(b) 0
(c) $$\frac { π }{4}$$
(d) $$\frac { π }{2}$$

Hint:
$$\vec a$$ = 8$$\vec b$$ and $$\vec c$$ = -7$$\vec b$$
Clearly $$\vec a$$ and $$\vec b$$ are parallel and $$\vec b$$ and $$\vec c$$ are anti-parallel.
∴ $$\vec a$$ and $$\vec c$$ are anti-parallel.
Hence, angle between $$\vec a$$ and $$\vec c$$ is π.

Question 18.
If the vectors $$\vec a$$ = $$\hat i$$ – $$\hat j$$ + 2$$\hat k$$, $$\vec b$$ = 2$$\hat i$$ + 4$$\hat j$$ + $$\hat k$$ and $$\vec c$$ = λ$$\hat i$$ + $$\hat j$$ + µ$$\hat k$$ are mutually orthogonal, then (λ, µ) =
(a) (-3, 2)
(b) (2, -3)
(c) (-2, 3)
(d)(3, -2).

Hint:
$$\vec a$$, $$\vec b$$ and $$\vec c$$ are mutually orthogonal
⇒ $$\vec b$$.$$\vec c$$ = 0 and $$\vec a$$.$$\vec c$$ = 0
⇒ (2$$\hat i$$ + 4$$\hat j$$ + $$\hat k$$). (λ$$\hat i$$ + $$\hat j$$ + µ$$\hat i$$) = 0
⇒ 2λ + 4 + µ = 0
⇒ 2λ + µ = -4 …………(1)
and ($$\hat i$$ – $$\hat j$$ + 2$$\hat k$$).(λ$$\hat i$$ + $$\hat j$$ + µ$$\hat k$$) = 0
⇒ λ – 1 + 2µ = 0
⇒ λ + 2µ = 1 ………….. (2)
Solving (1) and (2),
λ = -3 and µ = 2.

Question 19.
If (2$$\hat i$$ + 6$$\hat j$$ + 27$$\hat k$$) × ($$\hat i$$ + p$$\hat j$$ + q$$\hat k$$) = $$\vec 0$$, then the values ofp and q are?
(a) p = 6, q = 27
(b) p = 3, q = $$\frac { 27 }{2}$$
(c) p = 6, q = $$\frac { 27 }{2}$$
(d) p = 3, q = 27.

Answer: (b) p = 3, q = $$\frac { 27 }{2}$$
Hint:
(2$$\hat i$$ + 6$$\hat j$$ + 27$$\hat k$$) × ($$\hat i$$ + p$$\hat j$$ + q$$\hat k$$)
$$\left[\begin{array}{ccc} \hat{i} & \hat{j} & \hat{k} \\ 2 & 6 & 27 \\ 1 & p & q \end{array}\right]$$
By the question,
$$\hat i$$ (6q – 27p) –$$\hat j$$ (2q – 27) +$$\hat k$$ (2p – 6) = $$\vec 0$$
⇒ 6q – 27p = 0 ⇒ 2q – 9p = 0
2q – 27 = 0 ⇒ q = $$\frac { 27 }{2}$$
and 2p – 6 = 0 ⇒ p = 3.
Hence, p = 3 and q = $$\frac { 27 }{2}$$.

Question 20.
If the vectors $$\bar { AB }$$ = 3$$\hat i$$ + 4$$\hat k$$ and $$\bar { AC }$$ = 5$$\hat i$$ – 2$$\hat j$$ + 4$$\hat k$$ are the sides ofa triangle ABC, then the length of the median through A is
(a) $$\sqrt {72}$$
(b) $$\sqrt {33}$$
(c) $$\sqrt {45}$$
(d) $$\sqrt {18}$$

Answer: (b) $$\sqrt {33}$$
Hint:

Question 21.
If [$$\vec a$$ × $$\vec b$$ $$\vec b$$ × $$\vec c$$ $$\vec c$$ × $$\vec a$$] = λ [$$\vec a$$ $$\vec b$$ $$\vec c$$]², then λ is equal to
(a) 3
(b) 0
(c) 1
(d) 2.

Hint:
As usual, we will have:
[$$\vec a$$ × $$\vec b$$ $$\vec b$$ × $$\vec c$$ $$\vec c$$ × $$\vec a$$] = [$$\vec a$$ $$\vec b$$ $$\vec c$$]²
Given:
[$$\vec a$$ × $$\vec b$$ $$\vec b$$ × $$\vec c$$ $$\vec c$$ × $$\vec a$$] = [$$\vec a$$ $$\vec b$$ $$\vec c$$]²
Hence, λ = 1.

Question 22.
Let $$\vec a$$, $$\vec b$$ and $$\vec c$$ be three unit vectors such that:
$$\vec a$$ × ($$\vec b$$ × $$\vec c$$) = $$\frac { √3 }{2}$$ ($$\vec b$$ + $$\vec c$$)
If $$\vec b$$ is not parallel to $$\vec c$$, then the angle between $$\vec a$$ and $$\vec b$$ is:
(a) $$\frac { π }{2}$$
(b) $$\frac { 2π }{3}$$
(c) $$\frac { 5π }{6}$$
(d) $$\frac { 3π }{4}$$

Answer: (c) $$\frac { 5π }{6}$$
Hint:

Fill in the blanks

Question 1.
The magnitude of projection of (2$$\hat i$$ – $$\hat j$$ + $$\hat k$$)
on ($$\hat i$$ – 2$$\hat j$$ + 2$$\hat k$$) is ……………….

Question 2.
Vector of magnitude 5 units and in the direction opposite to 2$$\hat i$$ + 3$$\hat j$$ – 6$$\hat k$$ is ……………..

Answer: $$\frac { 5 }{7}$$ (-2$$\hat i$$ – 3$$\hat j$$ + 6$$\hat k$$)

Question 3.
The sum of the vectors
$$\vec a$$ = $$\hat i$$ – 2$$\hat j$$ + $$\hat k$$, $$\vec b$$ = -2$$\hat i$$ + 4$$\hat j$$ + 5$$\hat k$$ and $$\vec c$$ = $$\hat i$$ – 6$$\hat j$$ – 7$$\hat k$$ is ……………….

Answer: -4$$\hat i$$ – $$\hat k$$

Question 4.
The value of ‘a’ when the vectors:
2$$\hat i$$ – 3$$\hat j$$ + 4$$\hat k$$ and a$$\hat i$$ + b$$\hat j$$ – 8$$\hat k$$ are collinear is ……………….

Question 5.
If $$\vec a$$ = 2$$\hat i$$ + $$\hat j$$ – 2$$\hat k$$, then |$$\vec a$$| = ……………….

Question 6.
If $$\vec a$$ is a unit vector and ($$\vec x$$ – $$\vec a$$).($$\vec x$$ + $$\vec a$$) = 8, then |$$\vec x$$| = …………….

Question 7.
($$\hat i$$ × $$\hat j$$).$$\hat k$$ + $$\hat i$$.$$\hat j$$ = ……………..

Question 8.
The value of ‘λ’ of (2$$\hat i$$ + 6$$\hat j$$ + 14$$\hat k$$) × ($$\hat i$$ – λ$$\hat j$$ + 7$$\hat k$$) = $$\vec 0$$ is ……………….

Question 9.
If any two vectors $$\vec a$$, $$\vec b$$, $$\vec c$$ are parallel, then [$$\vec a$$. $$\vec b$$. $$\vec c$$] = …………………

Question 10.
The value of ‘λ’ such that the vectors:
3$$\hat i$$ + $$\hat j$$ + 5$$\hat k$$, $$\hat i$$ + 2$$\hat j$$ – 3$$\hat k$$ and 2$$\hat i$$ – $$\hat j$$ + $$\hat k$$ are coplanar is ………………..