The position vector of a point which divides the line joining the points with position vectors (a - 2b) and (2a + b) externally in the ratio 2:1, is Option 1: 3a + 4b Option 2: -3a + 4b Option 3: 3a - 4b Option 4: 5/3a

Answer: 3a + 4b. Solution: When a point divides the line joining points with position vectors p and q externally in the ratio m : n, its position vector is: r = (m * q - n * p)/(m - n) Here, m = 2, n = 1, p = a - 2b, q = 2a + b r = (2(2a + b) - 1(a - 2b))/(2 - 1) = (4a + 2b - a + 2b)/1 = (4a - a) + (2b + 2b) = 3a + 4b The position vector of the required point is 3a + 4b.

Algebra

Vector Algebra

Medium

core

The position vector of a point which divides the line joining the points with position vectors $(\vec{a} - 2\vec{b})$ and $(2\vec{a} + \vec{b})$ externally in the ratio 2:1, is

$3\vec{a} + 4\vec{b}$

$-3\vec{a} + 4\vec{b}$

$3\vec{a} - 4\vec{b}$

$\frac{5}{3}\vec{a}$

✅ Correct Option: 1

When a point divides the line joining points with position vectors $\vec{p}$ and $\vec{q}$ externally in the ratio $m : n$ , its position vector is:

$\vec{r} = \dfrac{m \cdot \vec{q} - n \cdot \vec{p}}{m - n}$

Here, $m = 2$ , $n = 1$ , $\vec{p} = \vec{a} - 2\vec{b}$ , $\vec{q} = 2\vec{a} + \vec{b}$

$\vec{r} = \dfrac{2(2\vec{a} + \vec{b}) - 1(\vec{a} - 2\vec{b})}{2 - 1}$

$= \dfrac{4\vec{a} + 2\vec{b} - \vec{a} + 2\vec{b}}{1}$

$= (4\vec{a} - \vec{a}) + (2\vec{b} + 2\vec{b})$

$= 3\vec{a} + 4\vec{b}$

The position vector of the required point is $3\vec{a} + 4\vec{b}$ .

2025: 13 May Shift 1

The vector equation of the line passing through points $A(3,4,-7)$ and $B(1,-1,6)$ is

2025: 21 May Shift 1

Match List-I with List-II

List-I	List-II
(A) The vectors $\lambda\hat{i}$ $+ \hat{j} + 2\hat{k}$ and $\vec{i} + \lambda\hat{j} + \hat{k}$ are perpendicular if λ is equal to	(I) 1
(B) The vectors $3\hat{i} + 6\hat{j} - \hat{k}$ and $2\hat{i} + 4\hat{j} - \lambda\hat{k}$ are collinear if λ is equal to	(II) -1
(C) The number of vectors of unit-length which are perpendicular to both the vectors $\vec{a}$ = $\hat{i} + \hat{j} + 2\hat{k}$ and $\vec{b}$ = $3\hat{i} - \hat{j} + 5\hat{k}$ is	(III) 2/3
(D) If $\lvert \vec{a} \rvert = 1$ and $\vec{a} + \vec{b} = \vec{0}$ , then $\lvert \vec{b} \rvert$ is equal to	(IV) 2

Choose the correct answer from the options given below:

2026: 19 May Shift 2

If $\theta$ is angle between $\vec{a}$ & $\vec{b}$ , then for vectors $\vec{a}, \vec{b}$ and $\vec{c}$ ,

Match the LIST-I with LIST-II

	LIST-I		LIST-II
A.	$\vec{a} \times \vec{b} = \vec{a} \times \vec{c}, \vec{a} \neq 0, \vec{b} \neq \vec{c}$	I.	$\theta = \dfrac{\pi}{4}$
B.	$\vec{a} \times \vec{b} = 0$	II.	$\vec{a} \parallel (\vec{b} - \vec{c})$
C.	$\vert \vec{a}.\vec{b} \vert = \vert \vec{a} \times \vec{b} \vert$	III.	$\vec{b} = \vec{c}$
D.	$\vec{a}.\vec{b} = \vec{a}.\vec{c}$ and $\vec{a} \times \vec{b} = \vec{a} \times \vec{c}$ , $\vec{a} \neq 0$	IV.	$\vec{a} = 0, \vec{b} = 0$ or $\vec{a} \parallel \vec{b}$

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: 15 May Shift 1

If $|\vec a|=2, |\vec b|=7, \vec a \times \vec b = 3\hat i + 2\hat j + 6\hat k$ , then the angle between $\vec a$ and $\vec b$ is equal to

2025: 3 June Shift 1

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

The position vector of a point which divides the line joining the points with position vectors (a⃗−2b⃗)(\vec{a} - 2\vec{b})(a−2b) and (2a⃗+b⃗)(2\vec{a} + \vec{b})(2a+b) externally in the ratio 2:1, is

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CUET Mathematics Past Year Questions (Topic-Wise)

The position vector of a point which divides the line joining the points with position vectors $(\vec{a} - 2\vec{b})$ and $(2\vec{a} + \vec{b})$ externally in the ratio 2:1, is