A vector a of magnitude 3√2 making an angle of ()/3 with i, ()/4 with j and an actue angle with k, is Option 1: 3√2(1/√2i + 1/√2j + 1/2k) Option 2: 3√2(1/2i + 1/√2j + 1/2k) Option 3: 3√2(1/√2i + 1/2j + 1/2k) Option 4: 3√2(1/2i + 1/2j + 1/2k)

Answer: 3√2(1/2i + 1/√2j + 1/2k). Solution: If a vector makes angles, with i, j, k respectively, then direction cosines satisfy: ^2 + ^2 + ^2 = 1 Given: = ()/3, so = ()/3 = 1/2 = ()/4, so = ()/4 = 1/√2 = (acute), so = \ ? ^2 + ^2 + ^2 = 1 (1/2)^2 + (1/√2)^2 + ^2 = 1 1/4 + 1/2 + ^2 = 1 3/4 + ^2 = 1 ^2 = 1/4 = ±1/2 Since is acute, is positive: = 1/2 a =, a, (\ i + \ j + \ k) = 3√2(1/2i + 1/√2j + 1/2k) Therefore, a = 3√2(1/2i + 1/√2j + 1/2k)

Algebra

Vector Algebra

Medium

core

A vector $\vec{a}$ of magnitude $3\sqrt{2}$ making an angle of $\frac{\pi}{3}$ with $\hat{i}$ , $\frac{\pi}{4}$ with $\hat{j}$ and an actue angle $\theta$ with $\hat{k}$ , is

$3\sqrt{2}(\frac{1}{\sqrt{2}}\hat{i} + \frac{1}{\sqrt{2}}\hat{j} + \frac{1}{2}\hat{k})$

$3\sqrt{2}(\frac{1}{2}\hat{i} + \frac{1}{\sqrt{2}}\hat{j} + \frac{1}{2}\hat{k})$

$3\sqrt{2}(\frac{1}{\sqrt{2}}\hat{i} + \frac{1}{2}\hat{j} + \frac{1}{2}\hat{k})$

$3\sqrt{2}(\frac{1}{2}\hat{i} + \frac{1}{2}\hat{j} + \frac{1}{2}\hat{k})$

✅ Correct Option: 2

If a vector makes angles $\alpha$ , $\beta$ , $\gamma$ with $\hat{i}$ , $\hat{j}$ , $\hat{k}$ respectively, then direction cosines satisfy:

$\cos^2\alpha + \cos^2\beta + \cos^2\gamma = 1$

Given:

$\alpha = \dfrac{\pi}{3}$ , so $\cos\alpha = \cos\dfrac{\pi}{3} = \dfrac{1}{2}$

$\beta = \dfrac{\pi}{4}$ , so $\cos\beta = \cos\dfrac{\pi}{4} = \dfrac{1}{\sqrt{2}}$

$\gamma = \theta$ (acute), so $\cos\theta = \ ?$

$\cos^2\alpha + \cos^2\beta + \cos^2\gamma = 1$

$\left(\dfrac{1}{2}\right)^2 + \left(\dfrac{1}{\sqrt{2}}\right)^2 + \cos^2\theta = 1$

$\dfrac{1}{4} + \dfrac{1}{2} + \cos^2\theta = 1$

$\dfrac{3}{4} + \cos^2\theta = 1$

$\cos^2\theta = \dfrac{1}{4}$

$\cos\theta = \pm\dfrac{1}{2}$

Since $\theta$ is acute, $\cos\theta$ is positive:

$\cos\theta = \dfrac{1}{2}$

$\vec{a} = |\vec{a}|\left(\cos\alpha\ \hat{i} + \cos\beta\ \hat{j} + \cos\theta\ \hat{k}\right)$

$= 3\sqrt{2}\left(\dfrac{1}{2}\hat{i} + \dfrac{1}{\sqrt{2}}\hat{j} + \dfrac{1}{2}\hat{k}\right)$

Therefore, $\vec{a} = 3\sqrt{2}\left(\dfrac{1}{2}\hat{i} + \dfrac{1}{\sqrt{2}}\hat{j} + \dfrac{1}{2}\hat{k}\right)$

2025: 3 June Shift 1

Let $\vec{a} = 3\hat{i} + \hat{j} - 4\hat{k}$ and $\vec{b} = 6\hat{i} + 5\hat{j} - 2\hat{k}$ be two vectors. Then a vector perpendicular to $\vec{a}$ and $\vec{b}$ with magnitude 3 units is

2025: 3 June Shift 1

Let $\vec{a} = \hat{i} + 4\hat{j} + 2\hat{k}$ , $\vec{b} = 3\hat{i} - 2\hat{j} + 7\hat{k}$ and $\vec{c} = 2\hat{i} + \hat{j} + 4\hat{k}$ . A vector $\vec{d}$ which is perpendicular to both $\vec{a}$ and $\vec{b}$ , and $\vec{c} \cdot \vec{d} = 14$ , is:

2022: 7 Aug Shift 2

Match List - I with List - II.

	List - I (Two given vector)		List - II (Projection of $\vec{a}$ on $\vec{b}$ )
(A)	$\vec{a} = \hat{i} - \hat{j}$ , $\vec{b} = \hat{i} + \hat{j}$	(I)	$\frac{2}{\sqrt{5}}$
(B)	$\vec{a} = \hat{i} + \hat{j}$ , $\vec{b} = 2\hat{i} - \hat{k}$	(II)	0
(C)	$\vec{a} = \hat{j} + \hat{k}$ , $\vec{b} = \hat{i} + \hat{k}$	(III)	$\sqrt{2}$
(D)	$\vec{a} = 2\hat{i} + 3\hat{j}$ , $\vec{b} = \hat{i} - \hat{k}$	(IV)	$\frac{1}{\sqrt{2}}$

Choose the correct answer from the options given below :

2026: 19 May Shift 1

If $\vec{a}$ and $\vec{b}$ are two collinear vectors, then which of the following statements is/are incorrect?

(A) $\vec{b} = \lambda\vec{a}$ , for some scalar $\lambda$ .

(B) Always $\vec{a} = \pm\vec{b}$ .

(D) both vectors $\vec{a}$ and $\vec{b}$ always have same direction.

Choose the correct answer from the options given below:

2025: 30 May Shift 1

If $\vec{a} + \vec{b} + \vec{c} = \vec{0}$ and $|\vec{a}| = 5, |\vec{b}| = 3, |\vec{c}| = 7$ , then the acute angle between $\vec{a}$ and $\vec{b}$ is

2025: 2 June Shift 1

The value of $\lambda$ , for which the two vectors $2\hat{i} - \hat{j} + 2\hat{k}$ and $3\vec{i} + \lambda\vec{j} + \hat{k}$ are perpendicular, is:

A vector a⃗\vec{a}a of magnitude 323\sqrt{2}32​ making an angle of π3\frac{\pi}{3}3π​ with i^\hat{i}i^, π4\frac{\pi}{4}4π​ with j^\hat{j}j^​ and an actue angle θ\thetaθ with k^\hat{k}k^, is

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CUET Mathematics Past Year Questions (Topic-Wise)

A vector $\vec{a}$ of magnitude $3\sqrt{2}$ making an angle of $\frac{\pi}{3}$ with $\hat{i}$ , $\frac{\pi}{4}$ with $\hat{j}$ and an actue angle $\theta$ with $\hat{k}$ , is