CUET UG 2022 — Mathematics Algebra

2022mcqmedium

If $\vec{a} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k}$ and $\vec{b} = b_1\hat{i} + b_2\hat{j} + b_3\hat{k}$ are two non zero vectors inclined at an angle $\theta$ , then identify the correct option out of the given options.
(a) $\cos\theta = \frac{\vec{a} \cdot \vec{b}}{|\vec{a}| \cdot |\vec{b}|}$
(b) $\vec{a}$ and $\vec{b}$ are perpendicular, if $a_1 b_1 + a_2 b_2 + a_3 b_3 = 0$
(c) $\vec{a}$ and $\vec{b}$ are perpendicular, if $\frac{a_1}{b_1} = \frac{a_2}{b_2} \neq \frac{c_1}{c_2}$
(d) for $\theta = \pi$ , $\vec{a} \times \vec{b} = 0$
(e) $\cos\theta = \frac{|\vec{a} \times \vec{b}|}{|\vec{a}| \cdot |\vec{b}|}$
Choose the most appropriate answer from the options given below

Official previous-year question

Held on 30 Aug 2022 · Verified 13 Jul 2026.

Options

A
(a), (b) and (d) only
B
(a), (b) and (e) only
C
(b), (d) and (e) only
D
(a) and (b) only

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Solution

(a) Standard formula for angle: $\cos\theta = \frac{\vec{a}\cdot\vec{b}}{|\vec{a}||\vec{b}|}$ -- correct.

(b) Perpendicular iff $\vec{a}\cdot\vec{b}=0$ , i.e., $a_1b_1+a_2b_2+a_3b_3=0$ -- correct.

(d) For $\theta=\pi$ , $\sin\pi=0$ , so $\vec{a}\times\vec{b}=\vec{0}$ -- correct.

(e) Wrong: that ratio equals $\sin\theta$ , not $\cos\theta$ .

So (a), (b), (d) are correct.

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Solution

(a) Standard formula for angle: $\cos\theta = \frac{\vec{a}\cdot\vec{b}}{|\vec{a}||\vec{b}|}$ -- correct.

(b) Perpendicular iff $\vec{a}\cdot\vec{b}=0$ , i.e., $a_1b_1+a_2b_2+a_3b_3=0$ -- correct.

(d) For $\theta=\pi$ , $\sin\pi=0$ , so $\vec{a}\times\vec{b}=\vec{0}$ -- correct.

(e) Wrong: that ratio equals $\sin\theta$ , not $\cos\theta$ .

So (a), (b), (d) are correct.

CUET UG 2022 — Mathematics Algebra

Options

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Solution

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CUET UG Algebra in other years

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CUET UG 2022 — Mathematics Algebra

Options

Did you get this right?

Solution

Your note

CUET UG Algebra in other years

More CUET UG Algebra questions

Other Mathematics topics for CUET UG

Calculus

Applied-Mathematics

Statistics & Applications

Geometry

Arithmetic

Coordinate Geometry

Keep practicing Algebra