Question

Find the direction cosines of the line passing through two points P(-6,7,3) and Q(3,-2,5).

(a) –\(\frac{2}{\sqrt{166}},\frac{-9}{\sqrt{166}},\frac{2}{\sqrt{166}}\)

(b) –\(\frac{9}{\sqrt{166}},\frac{-7}{\sqrt{166}},\frac{2}{\sqrt{166}}\)

(c) –\(\frac{9}{\sqrt{66}},\frac{-9}{\sqrt{66}},\frac{2}{\sqrt{66}}\)

(d) –\(\frac{9}{\sqrt{166}},\frac{-9}{\sqrt{166}},\frac{2}{\sqrt{166}}\)

The question was posed to me during a job interview.

This key question is from Direction Cosines and Direction Ratios of a Line in portion Three Dimensional Geometry of Mathematics – Class 12

Answer 1

Right answer is (d) –\(\frac{9}{\sqrt{166}},\frac{-9}{\sqrt{166}},\frac{2}{\sqrt{166}}\)

The best explanation: The direction cosines of two lines passing through two points is given by:

\(\frac{x_2-x_1}{PQ},\frac{y_2-y_1}{PQ},\frac{z_2-z_1}{PQ}\)

and \(PQ=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}\)

In the given problem we have, P(-6,7,3) and Q(3,-2,5)

∴\(PQ=\sqrt{(3+6)^2+(-2-7)^2+(5-3)^2}\)

=\(\sqrt{81+81+4}=\sqrt{166}\)

Hence, the direction ratios are \(l=\frac{-6-3}{\sqrt{166}}=-\frac{9}{\sqrt{166}}\)

m=\(\frac{-2-7}{\sqrt{166}}=\frac{-9}{\sqrt{166}}\)

n=\(\frac{5-3}{\sqrt{166}}=\frac{2}{\sqrt{166}}\)