Name: Shortest Distance Between Skew Lines
Uploaded: 2019-07-19
Duration: 5 min 52 s
Description: In this video, let us learn about how to find the shortest distance between two skewed or non-coplanar lines. Peruse this tutorial video now!

Question

Find the image of the point

(1, 6, 3)

in the line

\frac{x}{1} = \frac{y - 1}{2} = \frac{z - 2}{3}

.

Accepted Answer

Given equation of the line:

$\frac{x}{1} = \frac{y - 1}{2} = \frac{z - 2}{3}$

From $P (1, 6, 3)$ draw $P M$ perpendicular to $A B$ and produce it to $P' (a, b, c)$ such that $M$ is the mid point of $P P'$ . And $P'$ is the image of $P$ in line $A B$ .

Equation of line $A B$ can be written as:

$\frac{x}{1} = \frac{y - 1}{2} = \frac{z - 2}{3} = r$

So, any point $M$ on $A B$ is $(r, 2 r + 1, 3 r + 2)$ .

Direction ratios of $A B$ are $(1, 2, 3)$

Therefore, direction ratios of $P M$ are $r - 1, 2 r - 5, 3 r - 1 .$

Now, as it is given that $P M$ is perpendicular to $A B$ , so according to perpendicular condition:

$1 (r - 1) + 2 (2 r - 5) + 3 (3 r - 1) = 0$

$\Rightarrow r - 1 + 4 r - 10 + 9 r - 3 = 0$

$\Rightarrow 14 r - 14 = 0$

$\Rightarrow 14 r = 14$

$\Rightarrow r = \frac{14}{14} = 1$

So, point $M$ is $(1, 2 (1) + 1, 3 (1) + 2)$ i.e $(1, 3, 5)$ .

As $M$ is the mid point of $P P' .$

$\Rightarrow \frac{1 + a}{2} = 1 \frac{6 + b}{2} = 3 \frac{3 + c}{2} = 5$

$\Rightarrow 1 + a = 2 6 + b = 6 3 + c = 10$

$\Rightarrow a = 2 - 1 b = 6 - 6 c = 10 - 3$

$\Rightarrow a = 1 b = 0 c = 7$

So, the required image i.e $P'$ is $(1, 0, 7)$ .

Find the image of the point $(1, 6, 3)$ in the line $\frac{x}{1} = \frac{y - 1}{2} = \frac{z - 2}{3}$ .

Important Questions on Straight Lines

Learn and Prepare for any exam you want !

Find the image of the point 1,6,3 in the line x1=y-12=z-23.

Important Questions on Straight Lines

Learn and Prepare for any exam you want !

Find the image of the point $(1, 6, 3)$ in the line $\frac{x}{1} = \frac{y - 1}{2} = \frac{z - 2}{3}$ .