Algebra

Vector Algebra

Medium

core

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

$\vec{x} = 3\vec{i} + 4\vec{j} - 7\vec{k} + \lambda(\vec{i} - \vec{j} + 6\vec{k})$

$\vec{x} = (3 - 2\lambda)\vec{i} + (4 - 5\lambda)\vec{j} + (-7 + 13\lambda)\vec{k}$

$\vec{x} = \vec{i} - \vec{j} + 6\vec{k} + \lambda(3\vec{i} + 4\vec{j} - 7\vec{k})$

$\vec{x} = (-2 + 3\lambda)\vec{i} + (-5 + 4\lambda)\vec{j} + (13 - 7\lambda)\vec{k}$

✅ Correct Option: 2

The vector equation of a line passing through two points is $\vec{x} = \vec{a} + \lambda\vec{d}$ , where $\vec{a}$ is a position vector of any point on the line, $\vec{d}$ is the direction vector, and $\lambda$ is a parameter.

The direction vector is found by subtracting the coordinates of the two points:

$\vec{d} = B - A$

$\vec{d} = (1-3)\vec{i} + (-1-4)\vec{j} + (6-(-7))\vec{k}$

$\vec{d} = -2\vec{i} - 5\vec{j} + 13\vec{k}$

Using point $A(3, 4, -7)$ as the position vector:

$\vec{x} = (3\vec{i} + 4\vec{j} - 7\vec{k}) + \lambda(-2\vec{i} - 5\vec{j} + 13\vec{k})$

Expanding the equation:

$\vec{x} = 3\vec{i} + 4\vec{j} - 7\vec{k} - 2\lambda\vec{i} - 5\lambda\vec{j} + 13\lambda\vec{k}$

Grouping terms:

$\vec{x} = (3 - 2\lambda)\vec{i} + (4 - 5\lambda)\vec{j} + (-7 + 13\lambda)\vec{k}$

Therefore, the vector equation of the line is $\vec{x} = (3 - 2\lambda)\vec{i} + (4 - 5\lambda)\vec{j} + (-7 + 13\lambda)\vec{k}$

2026: 19 May Shift 1

Let $\vec{a} = (\hat{i}+4\hat{j}+2\hat{k})$ , $\vec{b} = (3\hat{i}-2\hat{j}+7\hat{k})$ , $\vec{c} = (2\hat{i}-\hat{j}+4\hat{k})$ . A vector $\vec{d}$ , which is perpendicular to both $\vec{a}$ and $\vec{b}$ , such that $\vec{c}.\vec{d} = 18$ , is

2024: 16 May Shift 1

The unit vector perpendicular to each of the vectors $\vec{a}+\vec{b}$ and $\vec{a}-\vec{b}$ , where $\vec{a}=\hat{i}+\hat{j}+\hat{k}$ and $\vec{b}=\hat{i}+2 \hat{j}+3 \hat{k}$ , is :

2026: 29 May Shift 2

Let $\vec{a}, \vec{b}, \vec{c}$ are three vectors such that $|\vec{a}|=2, |\vec{b}|=1, |\vec{c}|=3$ . If projection of $\vec{a}$ on $\vec{b}$ is equal to projection of $\vec{c}$ on $\vec{b}$ and $\vec{a}\perp\vec{c}$ , then value of $|2\vec{a}+3\vec{b}-2\vec{c}|$ is

2025: 21 May Shift 2

If $\vec{a}$ and $\vec{b}$ are two non-zero vectors such that $|\vec{a} \cdot \vec{b}| = |\vec{a} \times \vec{b}|$ , then the angle $\theta$ between $\vec{a}$ and $\vec{b}$ is

2025: 29 May Shift 2

Match List-I with List-II

Let $\theta$ be the angle between the vectors $\vec{a}$ and $\vec{b}$ .

List-I	List-II
(A) $\vec{a} \cdot \vec{b}$	(I) $\dfrac{\vec{a} \cdot \vec{b}}{\vert \vec{b}\vert ^2} \vec{b}$
(B) $\vec{a} \times \vec{b}$	(II) $\vec{a} \cdot \vec{b} = 0$
(C) Projection vector of $\vec{a}$ on $\vec{b}$ ( $\ne{0}$ )	(III) $\vert \vec{a}\vert \vert \vec{b}\vert \sin \theta \, \hat{n}$ where $\hat{n}$ is a unit vector perpendicular to both $\vec{a}$ and $\vec{b}$
(D) $\vec{a}$ and $\vec{b}$ are orthogonal vectors	(IV) $\vert \vec{a}\vert \vert \vec{b}\vert \cos \theta$

2026: 26 May Shift 2

If $\vec{a} = 5\hat{i} - \hat{j} - 3\hat{k}$ and $\vec{b} = \hat{i} - 3\hat{j} - 5\hat{k}$ then $(\vec{a}+\vec{b}).(\vec{a}-\vec{b})$ is equal to:

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

Solution

Related questions:

Keyboard Shortcuts

CUET Mathematics Past Year Questions (Topic-Wise)

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