If a + b + c = 0 and, a, = 5, b, = 3, c, = 7, then the acute angle between a and b is Option 1: ()/4 Option 2: ()/2 Option 3: ()/6 Option 4: ()/3

Answer: ()/3. Solution: Given three vectors that add up to zero: a + b + c = 0, a, = 5, b, = 3, c, = 7 From a + b + c = 0: c = -(a + b) Taking magnitude on both sides:, c, =, a + b, c, ^2 =, a + b, ^2, c, ^2 =, a, ^2 + 2a * b +, b, ^2 Substituting the given values: 7^2 = 5^2 + 2a * b + 3^2 49 = 25 + 2a * b + 9 49 = 34 + 2a * b 2a * b = 15 a * b = 15/2 Using the dot product formula a * b =, a, b, : 15/2 = 5 x 3 x 15/2 = 15 = 1/2 = ()/3 Therefore, the acute angle between a and b is ()/3.

Algebra

Vector Algebra

Medium

core

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

$\frac{\pi}{4}$

$\frac{\pi}{2}$

$\frac{\pi}{6}$

$\frac{\pi}{3}$

✅ Correct Option: 4

Given three vectors that add up to zero:

$\vec{a} + \vec{b} + \vec{c} = \vec{0}$

$|\vec{a}| = 5$

$|\vec{b}| = 3$

$|\vec{c}| = 7$

From $\vec{a} + \vec{b} + \vec{c} = \vec{0}$ :

$\vec{c} = -(\vec{a} + \vec{b})$

Taking magnitude on both sides:

$|\vec{c}| = |\vec{a} + \vec{b}|$

$|\vec{c}|^2 = |\vec{a} + \vec{b}|^2$

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

Substituting the given values:

$7^2 = 5^2 + 2\vec{a} \cdot \vec{b} + 3^2$

$49 = 25 + 2\vec{a} \cdot \vec{b} + 9$

$49 = 34 + 2\vec{a} \cdot \vec{b}$

$2\vec{a} \cdot \vec{b} = 15$

$\vec{a} \cdot \vec{b} = \dfrac{15}{2}$

Using the dot product formula $\vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}|\cos\theta$ :

$\dfrac{15}{2} = 5 \times 3 \times \cos\theta$

$\dfrac{15}{2} = 15\cos\theta$

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

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

Therefore, the acute angle between $\vec{a}$ and $\vec{b}$ is $\dfrac{\pi}{3}$ .

2026: 19 May Shift 2

Let $\vec{a}$ and $\vec{b}$ be two vectors, such that $|\vec{a}| = 3, |\vec{b}| = 5$ and the angle between them is $60°$ , then the value of $|\vec{a} + \vec{b}|$ is

2025: 14 May Shift 1

If $\hat{i},\hat{j}$ and $\hat{k}$ are unit vectors along co-ordinates axes OX, OY and OZ respectively, then which of the following is/are true?

(A) $\hat{i} \times \hat{i} = \vec{0}$

(B) $\hat{i} \times \hat{k} = \hat{j}$

(D) $\hat{i} \cdot \hat{j} = 0$

Choose the correct answer from the options given below:

2022: 6 Aug Shift 2

Let $\vec{OA} = 2\hat{i} - \hat{j} + \hat{k}$ and $\vec{OB} = \hat{i} + \hat{j} - \hat{k}$ . Then

A. The magnitude of vector $\vec{OA}$ is 6

B. The magnitude of vector $\vec{OB}$ is $\sqrt{3}$

C. The vector $\vec{AB}$ is $(-\hat{i} + 2\hat{j} - 2\hat{k})$

D. $\vec{OA} \cdot \vec{OB} = 0$

E. $\vec{OA} \parallel \vec{OB}$

Choose the correct answer from the options given below:

2025: 14 May Shift 2

Let $\vec{a} = \hat{i} + \hat{j}$ , $\vec{b} = \hat{i} - \hat{j}$ and $\vec{c} = \hat{i} + \hat{j} + \hat{k}$ . If $\hat{m}$ is a unit vector perpendicular to both $\vec{a}$ and $\vec{b}$ , then $|\vec{c}.\hat{m}|$ is equal to

2025: 16 May Shift 1

Match List-I with List-II

List-I	List-II
(A) Angle between î and -ĵ is	(I) $\frac{\pi}{6}$
(B) Angle between 2î + k̂ and 10î + 5k̂ is	(II) $\frac{\pi}{4}$
(C) Angle between î and î + ĵ is	(III) 2π
(D) Angle between √3ĵ - k̂ and ĵ is	(IV) $\frac{\pi}{2}$

Choose the correct answer from the options given below:

2025: 14 May Shift 1

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

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

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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