First slide

Straight lines in 3D

Question

The foot of the perpendicular from $P (5, 7, 3)$ to the line which is passing through the points $A (12, 21, 10)$ , $B (9, 13, 15)$ is

Moderate

A

$(2, 19, 7)$
B

$(- 2, 19, 7)$
C

$(5, 7, 3)$
D

$(9, 13, 15)$

Solution

The equation of the line passing through the points $A (12, 21, 10)$ and $B (9, 13, 15)$ is
   $\frac{x - 12}{3} = \frac{y - 21}{8} = \frac{z - 10}{- 5}$
General point on the line is $Q (3 k + 12, 8 k + 21, - 5 k + 10)$
Suppose that $Q$ be the foot of the perpendicular to the $P (5, 7, 3)$ point on the above line $\bar{P Q}$
Hence, the line  is perpendicular to the above line
It implies that
             $\begin{matrix} (3 k + 7) \cdot 3 + (8 k + 14) \cdot 8 + (- 5 k + 7) \cdot (- 5) = 0 \\ 9 k + 21 + 64 k + 112 + 25 k - 35 = 0 \\ 98 k + 98 = 0 \\ k = - 1 \end{matrix}$
Therefore, the foot of the perpendicular is    $Q (9, 13, 15)$

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