The length of the perpendicular drawn from the point (3,-1,11) to the line x2=y-23=z-34 is…

Question

The length of the perpendicular drawn from the point $(3,-1,11)$ to the line $\dfrac x2=\dfrac{y-2}3=\dfrac{z-3}4$ is:

Accepted Answer

Point $(2\alpha,3\alpha+2,4\alpha+3)$; direction of perpendicular $(2\alpha-3,3\alpha+3,4\alpha-8)\perp(2,3,4)$:
$$2(2\alpha-3)+3(3\alpha+3)+4(4\alpha-8)=0\implies 29\alpha-29=0\implies\alpha=1.$$
Foot $=(2,5,7)$; length $=\sqrt{1^2+6^2+4^2}=\sqrt{53}$.

Solutions · 1