First slide

Straight lines in 3D

Question

The foot of the perpendicular from $(2, - 4, 0)$ to the join of $(0, 0, 0), (0, 3, 0)$ is

Moderate

A

$(\frac{4}{3}, 0, 3)$
B

$(- \frac{4}{3}, 0, 0)$
C

$(0, - 4, 0)$
D

$(0, - \frac{4}{3}, 0)$

Solution

Equation of the line joining $(0, 0, 0), (0, 3, 0)$ is $\frac{x}{0} = \frac{y}{3} = \frac{z}{0}$
General point on the above line is $Q (0, 3 y_{1}, 0)$
The direction ratios of the line joining $P (2, - 4, 0)$ and $Q (0, 3 y_{1}, 0)$ is $〈- 2, 3 y_{1} + 4, 0〉$
Since the line $P Q$ and $\frac{x}{0} = \frac{y}{3} = \frac{z}{0}$ are perpendicular to each other.
$\begin{matrix} 3 (3 y_{1} + 4) = 0 \\ y_{1} = - \frac{4}{3} \end{matrix}$
Therefore, the foot of the perpendicular of $(2, - 4, 0)$ to the join of $(0, 0, 0), (0, 3, 0)$ is $Q (0, - 4, 0)$

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