Question

The coordinates of the foot of the perpendicular drawn from the point (0, 1, 2) on the x-axis are given by :

(A) (1, 0, 0)

(B) (2, 0, 0)

(C) (√5, 0, 0)

(D) (0, 0, 0)

Answer 1

Correct option is (D) (0, 0, 0)

Vector equation for x-axis is \(\vec r = \lambda \hat i\) where \(x = \lambda\). Let a point \(P(\lambda, 0, 0)\) is on the x-axis perpendicular to the given point \(A(0, 1, 2)\) then \(\overrightarrow {AP}\) is perpendicular to \(\lambda \hat i\).

\(\overrightarrow {AP} . \lambda \hat i = 0\) where \(\overrightarrow {AP} =\lambda \hat i -\hat j - 2\hat k\)

\((\lambda \hat i - \hat j - 2\hat k).(\lambda \hat i + 0\hat j + 0\hat k) = 0\)

\(\lambda^2 = 0\)

\(\lambda = 0\), so the foot of the perpendicular drawn from the point \((0, 1, 2)\) is \((0, 0, 0)\).