The position vector of a point R which divides the line joining two points P and Q whose position vectors are i + 2j - k and -i + j + k respectively in the ratio 2 : 1 externally is : Option 1: -1/3i + 4/3j + 1/3k Option 2: -3i + 3k Option 3: 3i - 3k Option 4: 1/3i - 4/3j - 1/3k

Answer: -3i + 3k. Solution: External division formula: R=(mQ-nP)/(m-n) with m:n=2:1. R=(2(-i+j+k)-(i+2j-k))/1=-3i+0j+3k=-3i+3k.

Algebra

Vector Algebra

Medium

core

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 :

$-\frac{1}{3}\hat{i} + \frac{4}{3}\hat{j} + \frac{1}{3}\hat{k}$

$-3\hat{i} + 3\hat{k}$

$3\hat{i} - 3\hat{k}$

$\frac{1}{3}\hat{i} - \frac{4}{3}\hat{j} - \frac{1}{3}\hat{k}$

✅ Correct Option: 2

External division formula: $\vec{R}=\frac{m\vec{Q}-n\vec{P}}{m-n}$ with $m:n=2:1$ .

$\vec{R}=\frac{2(-\hat{i}+\hat{j}+\hat{k})-(\hat{i}+2\hat{j}-\hat{k})}{1}=-3\hat{i}+0\hat{j}+3\hat{k}=-3\hat{i}+3\hat{k}$ .

2025: 26 May Shift 2

If $\vec{a}$ , $\vec{b}$ and $\sqrt{3}\vec{a} + \vec{b}$ are unit vectors, then the angle between $\vec{a}$ and $\vec{b}$ is:

2022: 17 Aug Shift 2

Match List - I with List - II.

List - I	List - II
(A) The value of $\hat{i}\cdot(\hat{j}\times\hat{k}) + \hat{j}\cdot(\hat{i}\times\hat{k}) + \hat{k}\cdot(\hat{i}\times\hat{j})$	(I) 16
(B) If $\lvert\vec{a}\rvert=10$ , $\lvert\vec{b}\rvert=2$ and $\vec{a}\cdot\vec{b}=12$ , then the value of $\lvert\vec{a}\times\vec{b}\rvert$ is	(II) $\frac{\pi}{4}$
(C) If $\theta$ is the angle between two vectors $\vec{a}$ and $\vec{b}$ , then the value of $\theta$ , for which $\vec{a}\cdot\vec{b} = \lvert\vec{a}\times\vec{b}\rvert$ is	(III) 14
(D) If $\vec{a}$ and $\vec{b}$ are perpendicular and $\vec{a} = 2\hat{i}+4\hat{j}+\lambda\hat{k}$ and $\vec{b} = 3\hat{i}-5\hat{j}+\hat{k}$ , then the value of $\lambda$ is	(IV) 1

Choose the correct answer from the options given below :

2026: 31 May Shift 2

If $|\vec{a}| = \sqrt{26}, |\vec{b}| = 7, |\vec{a}\times\vec{b}| = 35$ and $\theta$ be the angle between $\vec{a}$ and $\vec{b}$ then

Match the LIST-I with LIST-II

	LIST-I		LIST-II
A.	$\sin\theta$	I.	$61$
B.	$\cos\theta$	II.	$5/\sqrt{26}$
C.	$\vec{a} \cdot \vec{b}$	III.	$1/\sqrt{26}$
D.	$\vert \vec{a} - \vec{b} \vert^2$	IV.	$7$

Choose the correct answer from the options given below:

2026: 21 May Shift 1

Let us consider vector $\vec{a} = (c\log_2 x)\hat{i} - 6\hat{j} + 3\hat{k}$ and $\vec{b} = (\log_2 x)\hat{i} + 2\hat{j} + (2c\log_2 x)\hat{k}$ , then which of following statements are TRUE?

(Where $x \in \mathbb{R}$ (set of real numbers))

(A) Vector $\vec{a}$ and $\vec{b}$ make obtuse angle for any $x \in (0,\infty)$ if $c \in (-1,0)$

(B) Vector $\vec{a}$ and $\vec{b}$ makes obtuse angle for any $x \in \mathbb{R}$ if $c > 0$

(C) Vector $\vec{a}$ and $\vec{b}$ makes obtuse angle for any $x \in (0,\infty)$ if $c \in \left(-\frac{4}{3}, -1\right]$

(D) Vectors $\vec{a}$ and $\vec{b}$ makes obtuse angle for $x \in \mathbb{R}$ if $c \in (1,2)$

Choose the correct answer from the options given below:

2025: 19 May Shift 1

If $\vec{a} = 3\hat{i} - 6\vec{j} + \hat{k}$ and $\vec{b} = 2\hat{i} - 4\vec{j} + \lambda\hat{k}$ are such that $\vec{a} \parallel \vec{b}$ , then $3\lambda + 2 =$

2025: 3 June Shift 2

Let $\theta$ be the angle between two vectors $\vec{a}$ and $\vec{b}$ . Then match List-I with List-II

List-I	List-II
(A) $\sin \theta$	(I) $\dfrac{\vec{a} \cdot \vec{b}}{\vert \vec{a}\vert \vert \vec{b}\vert }$
(B) $\cos \theta$	(II) $\vert \vec{a} \times \vec{b}\vert$
(C) Area of the parallelogram with adjacent sides represented by $\vec{a}$ and $\vec{b}$	(III) $\dfrac{\vec{a} \cdot \vec{b}}{\vert \vec{a}\vert }$
(D) Projection of $\vec{a}$ on $\vec{b}$	(IV) $\dfrac{\vert \vec{a} \times \vec{b}\vert }{\vert \vec{a}\vert \vert \vec{b}\vert }$

Choose the correct answer from the options given below:

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

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CUET Mathematics Past Year Questions (Topic-Wise)

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 :