A perpendicular is drawn from a point $(1, 6, 3)$ to the line $\frac{x}{1} = \frac{y-2}{2} = \frac{z-3}{-3}$. What will be coordinates of the foot of perpendicular:
Step-by-Step Solution
Key Concept: The foot of perpendicular from a point to a line is found by setting the direction vector perpendicular to the line's direction.
Any point on line $\frac{x}{1} = \frac{y-1}{2} = \frac{z-2}{3} = λ$ is $P(λ, 2λ+1, 3λ+2)$. The direction vector of $AP$ is $⟨2λ-1, 2(2λ-5), 3(λ-1)⟩$. For perpendicularity to the line: $1(2λ-1) + 2(2λ-5) + 3(3λ-1) = 0$, giving $14λ-14 = 0$, so $λ = 1$. Therefore $P = (1, 3, 5)$.
Correct Answer: 1