First slide

Introduction to 3-D Geometry

Question

The foot of the perpendicular from $(5, 7, 3)$ to the join of $(9, 13, 15)$ , $(12, 21, 10)$ is

Moderate

A

$(- 2, - 19, 7)$
B

$(2, 19, 7)$
C

$(2, 2, 3)$
D

$(9, 13, 15)$

Solution

Let $A (5, 7, 3), B (9, 13, 15), C (12, 21, 10)$
Let P be the foot of the perpendicular from A to $\overset{\leftrightarrow}{B C}$ .
Let P divides $\overset{\leftrightarrow}{B C}$ in the ratio $k : 1$
$∴ P = (\frac{9 + 12 k}{1 + k}, \frac{13 + 21 k}{k + 1}, \frac{15 + 10 k}{1 + k})$
D.r’s of $\overset{\leftrightarrow}{A P}$ are $(\frac{9 + 12 k}{1 + k} - 5, \frac{13 + 21 k}{k + 1} - 7, \frac{15 + 10 k}{1 + k} - 3)$
D.r’s of $\overset{\leftrightarrow}{B C}$ are $(12 - 9, 21 - 13, 10 - 15) = (3, 8, - 5)$
$\overset{\leftrightarrow}{A P} ⊥ \overset{\leftrightarrow}{B C}$
$\Rightarrow 3 [\frac{9 + 12 k}{1 + k} - 5] + 8 [\frac{13 + 21 k}{1 + k} - 7]$
$\Rightarrow - 5 [\frac{15 + 10 k}{1 + k} - 3] = 0$
$∴$ Foot of the perpendicular $P = (9, 13, 15)$ ,

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