Name: Distance of a Point From a Plane
Uploaded: 2019-07-19
Duration: 9 min 54 s
Description: Three-dimensional space has an infinite number of points and planes in it! This video discusses the distance of a point from a plane, both in vector and cart...

Question

Find the coordinates of the foot of the perpendicular drawn from the point

(5, 4, 2)

to the line

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

Hence, or otherwise, deduce the length of the perpendicular.

Accepted Answer

Let $M$ be the foot of the perpendicular of the point $P (5, 4, 2)$ on the line $\frac{x + 1}{2} = \frac{y - 3}{3} = \frac{z - 1}{- 1}$

Therefore, its equation is $\frac{x + 1}{2} = \frac{y - 3}{3} = \frac{z - 1}{- 1} = r$

Then $M$ is in the form $(2 r - 1, 3 r + 3, - r + 1)$

We know that, if $A (x_{1}, y_{1}, z_{1}), B (x_{2}, y_{2}, z_{2})$ , then direction ratios of $A B =$ Position vector of $A$ $-$ Position vector of $B$ .

Direction ratios of $M P$ are $2 r - 1 - 5, 3 r + 3 - 4, - r + 1 - 2$ or $2 r - 6, 3 r - 1, - r - 1$

$\Rightarrow 2 r - 6, 3 r - 1, - r - 1$

Let $(a_{1}, b_{1}, c_{1}) = (2 r - 6, 3 r - 1, - r - 1)$

Since, $M P$ is perpendicular to the given line, dot product of direction ratios of their vector is equal to zero. Let direction ratios of line be $(a_{2}, b_{2}, c_{2}) = (2, 3, - 1)$ .

$\Rightarrow 2 (2 r - 6) + 3 (3 r - 1) - 1 (- r - 1) = 0$ Because $(a_{1} a_{2} + b_{1} b_{2} + c_{1} c_{2} = 0)$

$\Rightarrow 4 r - 12 + 9 r - 3 + r + 1 = 0$

$\Rightarrow 14 r - 14 = 0$

$\Rightarrow r = 1$

So, $M = (2 r - 1, 3 r + 3, - r + 1) = (2 (1) - 1, 3 (1) + 3, - 1 + 1) = (1, 6, 0)$

Length of the perpendicular $=$ distance between the coordinates $M (1, 6, 0)$ and $P (5, 4, 2)$

$MP = \sqrt{{(1 - 5)}^{2} + {(6 - 4)}^{2} + {(0 - 2)}^{2}} = \sqrt{16 + 4 + 4} = \sqrt{24} = 2 \sqrt{6}$ Units.

Find the coordinates of the foot of the perpendicular drawn from the point $(5, 4, 2)$ to the line $\frac{x + 1}{2} = \frac{y - 3}{3} = \frac{z - 1}{- 1} .$ Hence, or otherwise, deduce the length of the perpendicular.

Important Questions on The Plane

Learn and Prepare for any exam you want !

Find the coordinates of the foot of the perpendicular drawn from the point (5, 4, 2) to the line x+12=y-33=z-1-1. Hence, or otherwise, deduce the length of the perpendicular.

Important Questions on The Plane

Learn and Prepare for any exam you want !

Find the coordinates of the foot of the perpendicular drawn from the point $(5, 4, 2)$ to the line $\frac{x + 1}{2} = \frac{y - 3}{3} = \frac{z - 1}{- 1} .$ Hence, or otherwise, deduce the length of the perpendicular.