Vector Algebra MCQ with Solutions

Vector Algebra: Chapter 10 Links

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10.1 | 10.2 | 10.3 | 10.4 | Misc.

Vector Algebra MCQ with answers and solutions from NCERT Class 12 Maths. Revise all 12 important questions step by step with clear explanations. You can also watch the video solutions for better understanding. This is in continuation of the previous topic on Differential Equations MCQs.

Ready? Okay, let’s begin with the first question!

Vector Algebra MCQ#1 (NCERT Exercise 10.1 – Question 2 & 3)

📍Classify the following measures as scalars and vectors:
(i) $10\ kg$
(ii) $2\ meters\ north\text{-}west$
(iii) $40^\circ$
(iv) $40\ watt$
(v) $10^{-19}\ coulomb$
(vi) $20\ m/s^{2}$

Answer:
Scalars: (i), (iii), (iv), (v)
Vectors: (ii), (vi)

Explanation:
A scalar has only magnitude, whereas a vector has both magnitude and direction.

(i) $10\ kg$ → Scalar (mass has only magnitude)
(ii) $2\ meters\ north\text{-}west$ → Vector (magnitude + direction)
(iii) $40^\circ$ → Scalar (only magnitude of angle)
(iv) $40\ watt$ → Scalar (power is scalar)
(v) $10^{-19}\ coulomb$ → Scalar (electric charge is scalar)
(vi) $20\ m/s^{2}$ → Vector (acceleration has direction)

📍Classify the following as scalar and vector quantities:
(i) time period
(ii) distance
(iii) force
(iv) velocity
(v) work done

Answer:
Scalars: (i), (ii), (v)
Vectors: (iii), (iv)

Explanation:
(i) Time period → Scalar (only magnitude)
(ii) Distance → Scalar (no direction)
(iii) Force → Vector (magnitude + direction)
(iv) Velocity → Vector (magnitude + direction)
(v) Work done → Scalar (dot product of force and displacement gives scalar)

Watch Video Solution on YouTube

Vector Algebra MCQ#2 (NCERT Exercise 10.1 – Question 4)

📍In the given figure below, identify the following vectors:
(i) Coinitial
(ii) Equal
(iii) Collinear but not equal

Answer: (i) $\vec{a} \text{ and } \vec{d}$ (ii) $\vec{b} \text{ and } \vec{d}$ (iii) $\vec{a} \text{ and } \vec{c}$

Explanation:
Let the sides of the square be represented by $\vec{a}, \vec{b}, \vec{c}, \vec{d}$ in order.

(i) Coinitial vectors:
Vectors having the same initial point are called coinitial vectors.
From the figure, $\vec{a} \text{ and } \vec{d}$ start from the same vertex.

(ii) Equal vectors:
Equal vectors have same magnitude and same direction.
From the figure, $\vec{b} \text{ and } \vec{d}$ are equal vectors since they are parallel, equal in length and have the same direction.

(iii) Collinear but not equal:
Collinear vectors lie along the same straight line.
$\vec{a} \text{ and } \vec{c}$ are collinear but not equal (opposite direction).

Do watch the video below for better conceptual clarity.

Watch Video Solution on YouTube

Vector Algebra MCQ#3 (NCERT Exercise 10.1 – Question 5)

📍Answer the following as True or False:
(i) $\vec{a} \text{ and } -\vec{a}$ are collinear.
(ii) Two collinear vectors are always equal in magnitude.
(iii) Two vectors having same magnitude are collinear.
(iv) Two collinear vectors having the same magnitude are equal.

Answer:
(i) True
(ii) False
(iii) False
(iv) False

Explanation:
(i) $\vec{a} \text{ and } -\vec{a}$ lie along the same line but in opposite directions. Hence, they are collinear. ✔

(ii) Collinear vectors may have different magnitudes. Hence, this statement is False. ✘

(iii) Having same magnitude does not guarantee same direction. Hence, they need not be collinear. ✘

(iv) For vectors to be equal, both magnitude and direction must be same. Collinear vectors may have opposite directions. Hence, False. ✘

Do watch the video below for detailed conceptual understanding.

Watch Video Solution on YouTube

Let me tell you all of these 12 Vector Algebra MCQ are very simple and easy to understand and therefore scoring too. But you must practice them regularly.

So, my suggestion to you is to solve these Vector Algebra MCQ from NCERT a good number of times and feel confident. And if you ever need any clarification, just drop a comment — I’ll be more than happy to help.

Vector Algebra MCQ#4 (NCERT Exercise 10.2 – Question 19)

📍If $\vec{a}$ and $\vec{b}$ are two collinear vectors, then which of the following are incorrect:

(A) $\vec{b} = \lambda \vec{a}, \text{ for some scalar } \lambda$
(B) $\vec{a} = \pm \vec{b}$
(C) The respective components of $\vec{a}$ and $\vec{b}$ are not proportional
(D) Both the vectors $\vec{a}$ and $\vec{b}$ have same direction, but different magnitudes.

Answer:
✅ Correct option: (B), (C), (D)

Explanation:
If two vectors are collinear, then one must be a scalar multiple of the other:

$\vec{b} = \lambda \vec{a}$ ✔

So option (A) is correct.

(B) $\vec{a} = \pm \vec{b}$ restricts the magnitudes to be equal. Collinear vectors need not have equal magnitudes. Hence incorrect. ❌

(C) For collinear vectors, components must be proportional. So saying they are not proportional is incorrect. ❌

(D) Collinear vectors may have opposite directions.
When $\lambda$ is negative, the vectors are in opposite directions.
Hence, this statement is incorrect. ❌

Video Solution on YouTube

Vector Algebra MCQ#5 (NCERT Exercise 10.2 – Question 18)

📍In the given triangle ABC, which of the following is not true:

(A) $\overrightarrow{AB} + \overrightarrow{BC} + \overrightarrow{CA} = \vec{0}$
(B) $\overrightarrow{AB} + \overrightarrow{BC} - \overrightarrow{AC} = \vec{0}$
(C) $\overrightarrow{AB} + \overrightarrow{BC} + \overrightarrow{AC} = \vec{0}$
(D) $\overrightarrow{AB} - \overrightarrow{CB} + \overrightarrow{CA} = \vec{0}$

Answer:
✅ Correct option: (C)

Explanation:
By triangle law of vector addition:

$\overrightarrow{AB} + \overrightarrow{BC} = \overrightarrow{AC}$

Rearranging,
$\overrightarrow{AB} + \overrightarrow{BC} - \overrightarrow{AC} = \vec{0}$ ✔

Also, traversing the triangle in order gives:
$\overrightarrow{AB} + \overrightarrow{BC} + \overrightarrow{CA} = \vec{0}$ ✔

Further, since $-\overrightarrow{CB} = \overrightarrow{BC}$ , option (D) becomes:

$\overrightarrow{AB} + \overrightarrow{BC} + \overrightarrow{CA} = \vec{0}$ ✔

But in option (C), we have:

$\overrightarrow{AB} + \overrightarrow{BC} + \overrightarrow{AC}$

Since $\overrightarrow{AB} + \overrightarrow{BC} = \overrightarrow{AC}$ , this becomes:

$\overrightarrow{AC} + \overrightarrow{AC} = 2\overrightarrow{AC} \neq \vec{0}$

Hence, option (C) is not true.

Watch Video Solution on YouTube

Following is the next Vector Algebra MCQ in this part of the series.

Vectors MCQ#6 (NCERT Exercise 10.3 – Question 18)

📍If $\vec{a}$ is a non-zero vector of magnitude $a$ and $\lambda$ a non-zero scalar, then $\lambda \vec{a}$ is a unit vector if

(A) $\lambda = 1$
(B) $\lambda = -1$
(C) $a = |\lambda|$
(D) $a = \dfrac{1}{|\lambda|}$

Answer:
✅ Correct option: (D)

Explanation:
Magnitude of $\lambda \vec{a}$ is

$|\lambda \vec{a}| = |\lambda|\,|\vec{a}|$

Since $|\vec{a}| = a$ , we get

$|\lambda \vec{a}| = |\lambda| a$

For $\lambda \vec{a}$ to be a unit vector, its magnitude must be 1.

So,
$|\lambda| a = 1$

$a = \dfrac{1}{|\lambda|}$

Hence, option (D) is correct.

Watch Video Solution on Yoube

Did you know, there are about 100 MCQs in NCERT textbooks for Class 12 (both Part 1 and Part 2)? I shall be taking one Chapter at a time and show you the explanations for each of the questions.

Moreover, I have also covered these solutions in the video form and they are freely available on my YouTube channel, @Mathsbetter. And here also, in each of the 12 Vector Algebra MCQ on this page, I have provided the direct links to each and every MCQ for you to understand the solutions in a very simple and easy manner.

Vector Algebra MCQ#7 (NCERT Exercise 10.4 – Question 11)

📍Let the vectors $\vec{a}$ and $\vec{b}$ be such that $|\vec{a}| = 3$ and $|\vec{b}| = \dfrac{\sqrt{2}}{3}$ , then $\vec{a} \times \vec{b}$ is a unit vector, if the angle between $\vec{a}$ and $\vec{b}$ is

(A) $\dfrac{\pi}{6}$
(B) $\dfrac{\pi}{4}$
(C) $\dfrac{\pi}{3}$
(D) $\dfrac{\pi}{2}$

Answer:
✅ Correct option: (B)

Explanation:
Magnitude of cross product:

$|\vec{a} \times \vec{b}| = |\vec{a}|\,|\vec{b}| \sin\theta$

Substituting values:

$= 3 \cdot \dfrac{\sqrt{2}}{3} \sin\theta$

$= \sqrt{2}\sin\theta$

Since it is a unit vector:

$\sqrt{2}\sin\theta = 1$

i.e. $\sin\theta = \dfrac{1}{\sqrt{2}}$

$\theta = \dfrac{\pi}{4}$

Hence, correct option is (B).

Watch Video Solution on YouTube

Vector Algebra MCQ#8 (NCERT Exercise 10.4 – Question 12)

📍Area of a rectangle having vertices A, B, C and D with position vectors $-\hat{i} + \dfrac{1}{2}\hat{j} + 4\hat{k}, \hat{i} + \dfrac{1}{2}\hat{j} + 4\hat{k}, \hat{i} - \dfrac{1}{2}\hat{j} + 4\hat{k} \text{ and } -\hat{i} - \dfrac{1}{2}\hat{j} + 4\hat{k}$ , respectively is

(A) $\dfrac{1}{2}$
(B) $1$
(C) $2$
(D) $4$

Answer:
✅ Correct option: (C)

Explanation:
Let us find two adjacent sides.

$\vec{AB} = \vec{OB} - \vec{OA} = (2\hat{i})$

$\vec{BC} = \vec{OC} - \vec{OB} = (-\hat{j})$

Area of rectangle = $|\vec{AB} \times \vec{BC}|$

$= |2\hat{i} \times (-\hat{j})|$

$= | -2\hat{k} | = 2$

Hence, area = $2$ .

Watch Video Solution on YouTube

You may have been following this website as a resource for some of the important questions. For example, how to Integrate Square Root of tan x, or may be a complete guide to look at using Matrix Method in solving a system of linear equations.

Then trust me, similarly, this series on NCERT Class 12 Vector Algebra MCQ is going to help you in many ways.

Vectors MCQ#9 (NCERT Misc. Exercise Chapter 10 – Question 16)

📍If $\theta$ is the angle between two vectors $\vec{a}$ and $\vec{b}$ , then $\vec{a} \cdot \vec{b} \ge 0$ only when

(A) $\displaystyle 0 < \theta < \frac{\pi}{2}$
(B) $\displaystyle 0 \le \theta \le \frac{\pi}{2}$
(C) $\displaystyle 0 < \theta < \pi$
(D) $\displaystyle 0 \le \theta \le \pi$

Answer:
✅ Correct option: (B)

Explanation:
Since $\vec{a}\cdot\vec{b} = |\vec{a}||\vec{b}|\cos\theta$ and $\cos\theta \ge 0$ when $0 \le \theta \le \dfrac{\pi}{2}$ .

Watch Video Solution on YouTube

Vector Algebra MCQ#10 (NCERT Misc. Exercise Chapter 10 – Question 17)

📍Let $\vec{a}$ and $\vec{b}$ be two unit vectors and $\theta$ is the angle between them. Then $\vec{a} + \vec{b}$ is a unit vector if

(A) $\theta = \dfrac{\pi}{4}$
(B) $\theta = \dfrac{\pi}{3}$
(C) $\theta = \dfrac{\pi}{2}$
(D) $\theta = \dfrac{2\pi}{3}$

Answer:
✅ Correct option: (D)

Explanation:
$|\vec{a}+\vec{b}|^2 = |\vec{a}|^2 + |\vec{b}|^2 + 2\vec{a}\cdot\vec{b}$

$= 1 + 1 + 2\cos\theta = 2 + 2\cos\theta$

For unit vector:

$2 + 2\cos\theta = 1$

i.e. $\cos\theta = -\dfrac{1}{2}$

$\theta = \dfrac{2\pi}{3}$ .

Watch Video Solution on YouTube

Vector Algebra MCQ#11 (NCERT Misc. Exercise Chapter 10 – Question 18)

📍The value of $\hat{i}\cdot(\hat{j}\times\hat{k}) + \hat{j}\cdot(\hat{i}\times\hat{k}) + \hat{k}\cdot(\hat{i}\times\hat{j})$ is

(A) 0
(B) −1
(C) 1
(D) 3

Answer:
✅ Correct option: (C)

Explanation:
We have, $\hat{i}\cdot(\hat{j}\times\hat{k}) = \hat{i}\cdot\hat{i} = 1$
Similarly, $\hat{j}\cdot(\hat{i}\times\hat{k}) = -1$
And $\hat{k}\cdot(\hat{i}\times\hat{j}) = 1$

Total = $1 - 1 + 1 = 1$ .

Watch Video Solution on YouTube

Vector Algebra MCQ#12 (NCERT Misc. Exercise Chapter 10 – Question 19)

📍If $\theta$ is the angle between any two vectors $\vec{a}$ and $\vec{b}$ , then $|\vec{a}\cdot\vec{b}| = |\vec{a}\times\vec{b}|$ when $\theta$ is equal to

(A) 0
(B) $\dfrac{\pi}{4}$
(C) $\dfrac{\pi}{2}$
(D) $\pi$

Answer:
✅ Correct option: (B)

Explanation:
$|\vec{a}\cdot\vec{b}| = |\vec{a}||\vec{b}||\cos\theta|$
$|\vec{a}\times\vec{b}| = |\vec{a}||\vec{b}||\sin\theta|$

Equating:

$|\cos\theta| = |\sin\theta|$

$\theta = \dfrac{\pi}{4}$ .

Watch Video Solution on YouTube

Quick Answer Key

Q1 → (A) Scalars: (i), (iii), (iv), (v)
Vectors: (ii), (vi) (B) Scalars: (i), (ii), (v)
Vectors: (iii), (iv)
Q2 →(i) $\vec{a} \text{ and } \vec{d}$ (ii) $\vec{b} \text{ and } \vec{d}$ (iii) $\vec{a} \text{ and } \vec{c}$
Q3 →(i) True
(ii) False
(iii) False
(iv) False
Q4 → (B), (C), (D)
Q5 → (C)
Q6 → (D)
Q7 → (B)
Q8 → (C)
Q9 → (B)
Q10 → (D)
Q11 → (C)
Q12 → (B)

Closing Note

That completes all the important NCERT Class 12 Maths Vector Algebra MCQ from chapter 10.
I hope the explanations and video solutions made things clearer and gave you more confidence. Stay tuned for the next chapter, where we’ll cover all the important Probability MCQs (Chapter 13 from NCERT) with the same clarity and video support.

👉 Make sure to practice these questions again to strengthen your concepts.
👉 Watch the video solutions for a clearer and faster revision.

Keep practicing, and you’ll master the vector algebra easily! 🚀

Vector Algebra: Chapter 10 Links

Vector Algebra MCQ#1 (NCERT Exercise 10.1 – Question 2 & 3)

Vector Algebra MCQ#2 (NCERT Exercise 10.1 – Question 4)

Vector Algebra MCQ#3 (NCERT Exercise 10.1 – Question 5)

Vector Algebra MCQ#4 (NCERT Exercise 10.2 – Question 19)

Vector Algebra MCQ#5 (NCERT Exercise 10.2 – Question 18)

Vectors MCQ#6 (NCERT Exercise 10.3 – Question 18)

Vector Algebra MCQ#7 (NCERT Exercise 10.4 – Question 11)

Vector Algebra MCQ#8 (NCERT Exercise 10.4 – Question 12)

Vectors MCQ#9 (NCERT Misc. Exercise Chapter 10 – Question 16)

Vector Algebra MCQ#10 (NCERT Misc. Exercise Chapter 10 – Question 17)

Vector Algebra MCQ#11 (NCERT Misc. Exercise Chapter 10 – Question 18)

Vector Algebra MCQ#12 (NCERT Misc. Exercise Chapter 10 – Question 19)

Quick Answer Key

Closing Note

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