If, a - r, =, a, =, r, = 1, then angle between a and r is Option 1: ()/3 Option 2: (3)/4 Option 3: ()/2 Option 4: ()/6

Answer: ()/3. Solution: Given, a - r, = 1, a, = 1, and, r, = 1. The magnitude of the difference between two vectors can be expressed as:, a - r, ^2 =, a, ^2 +, r, ^2 - 2, a, r, where is the angle between the vectors. Substituting the given values: 1^2 = 1^2 + 1^2 - 2(1)(1) 1 = 1 + 1 - 2 1 = 2 - 2 2 = 2 - 1 2 = 1 = 1/2 The angle whose cosine equals 1/2 is: = ()/3 Therefore, the angle between a and r is ()/3.

Algebra

Vector Algebra

Medium

core

If $|\vec{a} - \vec{r}| = |\vec{a}| = |\vec{r}| = 1$ , then angle between $\vec{a}$ and $\vec{r}$ is

$\frac{\pi}{3}$

$\frac{3\pi}{4}$

$\frac{\pi}{2}$

$\frac{\pi}{6}$

✅ Correct Option: 1

Given $|\vec{a} - \vec{r}| = 1$ , $|\vec{a}| = 1$ , and $|\vec{r}| = 1$ .

The magnitude of the difference between two vectors can be expressed as:

$|\vec{a} - \vec{r}|^2 = |\vec{a}|^2 + |\vec{r}|^2 - 2|\vec{a}||\vec{r}|\cos\theta$

where $\theta$ is the angle between the vectors.

Substituting the given values:

$1^2 = 1^2 + 1^2 - 2(1)(1)\cos\theta$

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

$1 = 2 - 2\cos\theta$

$2\cos\theta = 2 - 1$

$2\cos\theta = 1$

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

The angle whose cosine equals $\frac{1}{2}$ is:

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

Therefore, the angle between $\vec{a}$ and $\vec{r}$ is $\frac{\pi}{3}$ .

2025: 15 May Shift 2

If $\hat{a}, \hat{b}$ and $\hat{c}$ are three unit vectors and $\hat{a} + \hat{b} + \hat{c} = \vec{0}$ , then the angle between $\hat{a}$ and $(-\hat{b})$ is

2026: 25 May Shift 1

If $\lvert\vec{a}\rvert = 4, \lvert\vec{b}\rvert = 3$ and $\lvert\vec{a}+\vec{b}\rvert = 6$ , then $\lvert\vec{a}-\vec{b}\rvert =$

2025: 13 May Shift 1

Match List-I with List-II

List-I	List-II
(A) Angle between $\vec{i} - \vec{j}$ and $\vec{j} + \vec{k}$	(I) 0
(B) Angle between $2\vec{j} - \vec{k}$ and $\vec{j} + 2\vec{k}$	(II) $\frac{2\pi}{3}$
(C) Angle between $\vec{i} + 2\vec{j}$ and $5\vec{i} + 10\vec{j}$	(III) $\frac{\pi}{6}$
(D) Angle between $\sqrt{3}\vec{i} + \vec{j}$ and $\vec{i}$	(IV) $\frac{\pi}{2}$

Choose the correct answer from the options given below:

2025: 2 June Shift 1

Let $\vec{a}$ and $\vec{b}$ are unit vectors. If $\sqrt{3}\vec{a} - \vec{b}$ is a unit vector, then the angle between $\vec{a}$ and $\vec{b}$ is

2023: 22 May Shift 3

The position vector of a point R which divides the line joining two points P and Q whose position vectors are $\hat{i} + 2\hat{j} - \hat{k}$ and $-\hat{i} + \hat{j} + \hat{k}$ respectively in the ratio 2 : 1 externally is :

2025: 30 May Shift 2

The projection vector of the vector $2\hat{i} + 3\hat{j} + \hat{k}$ on $2\hat{i} + \hat{j} - 2\hat{k}$ is

If ∣a⃗−r⃗∣=∣a⃗∣=∣r⃗∣=1|\vec{a} - \vec{r}| = |\vec{a}| = |\vec{r}| = 1∣a−r∣=∣a∣=∣r∣=1, then angle between a⃗\vec{a}a and r⃗\vec{r}r is

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CUET Mathematics Past Year Questions (Topic-Wise)

If $|\vec{a} - \vec{r}| = |\vec{a}| = |\vec{r}| = 1$ , then angle between $\vec{a}$ and $\vec{r}$ is