(B) The perpendicular distance $d$ from a point with position vector $c$ to the line $r = a + t\,b$ is given by the formula $d = \frac{|(c - a) \times

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(B) The perpendicular distance $d$ from a point with position vector $c$ to the line $r = a + t\,b$ is given by the formula $d = \frac{|(c - a) \times b|}{|b|}$.Let $P$ be any point on the line $r = a + t\,b$. The vector from the point $A$ (position vector $a$) to the point $C$ (position vector $c$) is $(c - a)$.The perpendicular distance $d$ is the magnitude of the component of the vector $(c - a)$ perpendicular to the vector $b$.This is given by the projection of $(c - a)$ onto the direction perpendicular to $b$,which is $\frac{|(c - a) \times b|}{|b|}$.

Class 12 Mathematics Vector Algebra Vector or Cross product of two vectors and its applications

Difficult

If the equation of a straight line passing through a point with position vector $a$ and parallel to the vector $b$ is $r = a + t\,b$,where $t$ is a parameter,then what is the perpendicular distance from this line to a point with position vector $c$?

A
$|(c - a) \times b| \div |a|$
B
$|(c - a) \times b| \div |b|$
C
$|(a - b) \times c| \div |c|$
D
$|(a - b) \times c| \div |a + c|$

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Vector or Cross product of two vectors and its applications

If $3\hat{i} + 4\hat{j}$ and $-5\hat{i} + 7\hat{j}$ are the vector sides of a triangle,then its area is given by

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Let $\overrightarrow{a}=\hat{i}+2 \hat{j}+3 \hat{k}$ and $\overrightarrow{b}=\hat{i}-2 \hat{j}-3 \hat{k}$ be two vectors. If $A_1$ is the area of the quadrilateral having $\vec{a}, \vec{b}$ as its diagonals and $A_2$ is the area of the parallelogram having $\overrightarrow{a}, \overrightarrow{b}$ as its two adjacent sides,then $A_1 \cdot A_2=$

EasyAP EAMCET 2023

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Let $\vec{a}, \vec{b}, \vec{c}$ be three vectors mutually perpendicular to each other and have the same magnitude. If a vector $\vec{r}$ satisfies $\vec{a} \times \{(\vec{r}-\vec{b}) \times \vec{a}\} + \vec{b} \times \{(\vec{r}-\vec{c}) \times \vec{b}\} + \vec{c} \times \{(\vec{r}-\vec{a}) \times \vec{c}\} = \vec{0}$,then $\vec{r}$ is equal to:

MediumJEE MAIN 2021

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Let $\vec{a} = 3\hat{i} + 2\hat{j} + x\hat{k}$ and $\vec{b} = \hat{i} - \hat{j} + \hat{k}$,for some real $x$. Then $|\vec{a} \times \vec{b}| = r$ is possible if

DifficultJEE MAIN 2019

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If $\vec{a}=\hat{i}+\hat{j}-\hat{k}$, $\vec{b}=\hat{i}-\hat{j}+\hat{k}$ and $\vec{c}$ is a unit vector perpendicular to $\vec{a}$ and coplanar with $\vec{a}$ and $\vec{b}$, then the unit vector $\vec{d}$ perpendicular to both $\vec{a}$ and $\vec{c}$ is

MediumWBJEE 2022

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If the equation of a straight line passing through a point with position vector $a$ and parallel to the vector $b$ is $r = a + t\,b$,where $t$ is a parameter,then what is the perpendicular distance from this line to a point with position vector $c$?

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If $3\hat{i} + 4\hat{j}$ and $-5\hat{i} + 7\hat{j}$ are the vector sides of a triangle,then its area is given by

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