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Question

Mathematics Question on 3D Geometry

The direction cosines of the line which is perpendicular to the lines with direction ratios 1,-2,-2 and 0, 2, 1 are:

A

23,−13,23\frac{2}{3}, -\frac{1}{3}, \frac{2}{3}

B

−23,−13,23-\frac{2}{3}, -\frac{1}{3}, \frac{2}{3}

C

23,−13,−23\frac{2}{3}, -\frac{1}{3}, -\frac{2}{3}

D

23,13,23\frac{2}{3}, \frac{1}{3}, \frac{2}{3}

Answer

23,−13,23\frac{2}{3}, -\frac{1}{3}, \frac{2}{3}

Explanation

Solution

Use the concept that the direction cosines of a line perpendicular to two given lines can be found by taking the cross product of the direction ratios.

Compute the cross product of ⟨1,−2,−2⟩\langle 1, -2, -2 \rangle and ⟨0,2,1⟩\langle 0, 2, 1 \rangle.

Normalize the resulting vector to get the direction cosines, confirming option (1) as the correct answer.