The distance between a point P whose position | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(D) Let the point $P$ be $(5, 1, 3)$.The line passes through point $Q(3, 7, 1)$ and is parallel to the vector $\vec{v} = \hat{j} + \hat{k}$.The vector

Vedclass · Accepted Answer

$6$

Vedclass · Answer

(D) Let the point $P$ be $(5, 1, 3)$.The line passes through point $Q(3, 7, 1)$ and is parallel to the vector $\vec{v} = \hat{j} + \hat{k}$.The vector $\vec{PQ} = (3-5)\hat{i} + (7-1)\hat{j} + (1-3)\hat{k} = -2\hat{i} + 6\hat{j} - 2\hat{k}$.The distance $d$ from a point to a line is given by $d = \frac{|\vec{PQ} 	imes \vec{v}|}{|\vec{v}|}$.First,calculate the cross product $\vec{PQ} 	imes \vec{v}$:$\vec{PQ} 	imes \vec{v} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \ -2 & 6 & -2 \ 0 & 1 & 1 \end{vmatrix} = \hat{i}(6 - (-2)) - \hat{j}(-2 - 0) + \hat{k}(-2 - 0) = 8\hat{i} + 2\hat{j} - 2\hat{k}$.The magnitude $|\vec{PQ} 	imes \vec{v}| = \sqrt{8^2 + 2^2 + (-2)^2} = \sqrt{64 + 4 + 4} = \sqrt{72} = 6\sqrt{2}$.The magnitude $|\vec{v}| = \sqrt{0^2 + 1^2 + 1^2} = \sqrt{2}$.Therefore,$d = \frac{6\sqrt{2}}{\sqrt{2}} = 6$.

The distance between a point $P$ whose position vector is $5 \hat{i}+\hat{j}+3 \hat{k}$ and the line $r=(3 \hat{i}+7 \hat{j}+\hat{k})+t(\hat{j}+\hat{k})$ is

Similar Questions

Point $A$ lies at a distance of $6$ units from the point $(1, 0, 1)$ on the line $\frac{x - 1}{2} = \frac{y}{2} = \frac{z - 1}{1}$ in the negative $z$-direction. Find the coordinates of $A$.

Find the angle between the following pair of lines:
$\frac{x-2}{2}=\frac{y-1}{5}=\frac{z+3}{-3}$ and $\frac{x+2}{-1}=\frac{y-4}{8}=\frac{z-5}{4}$

The angle between the lines whose direction cosines satisfy the equations $l+m+n=0$ and $l^2+m^2-n^2=0$ is

Let $L_1: \frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{4}$ and $L_2: \frac{x-2}{3}=\frac{y-4}{4}=\frac{z-5}{5}$ be two lines. Then which of the following points lies on the line of the shortest distance between $L_1$ and $L_2$?

The values of $p$ and $q$ such that the line joining the points $(7, p, 2)$ and $(q, -2, 5)$ is parallel to the line joining the points $(2, -3, 5)$ and $(-6, -15, 11)$ are:

Vedclass Test Series

Exam Paper Generator

Online Exam Module