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Question
Find the direction cosines of the line passing through two points (−2, 4, −5) and (1, 2, 3) .
Solution
\[\text{The direction cosines of the line passing through two points }P \left( x_1 , y_1 , z_1 \right) \text{ and} \ Q \left( x_2 , y_2 , z_2 \right) \text{are} \frac{x_2 - x_1}{PQ}, \frac{y_2 - y_1}{PQ}, \frac{z_2 - z_1}{PQ} . \]\[\text{ Here,} \]
\[PQ = \sqrt{\left( x_2 - x_1 \right)^2 + \left( y_2 - y_1 \right)^2 + \left( z_2 - z_1 \right)^2}\]
\[P = \left( - 2, 4, - 5 \right) \]
\[Q = \left( 1, 2, 3 \right)\]
\[ \therefore PQ = \sqrt{\left[ 1 - \left( - 2 \right) \right]^2 + \left( 2 - 4 \right)^2 + \left[ 3 - \left( - 5 \right) \right]^2} = \sqrt{77}\]
\[\text{Thus, the direction cosines of the line joining two points are }\frac{1 - \left( - 2 \right)}{\sqrt{77}}, \frac{2 - 4}{\sqrt{77}}, \frac{3 - \left( - 5 \right)}{\sqrt{77}}, \text{i . e }. \frac{3}{\sqrt{77}}, \frac{- 2}{\sqrt{77}}, \frac{8}{\sqrt{77}} .\]
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