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# Find the Direction Cosines of the Line Passing Through Two Points (−2, 4, −5) and (1, 2, 3) . - CBSE (Arts) Class 12 - Mathematics

ConceptDirection Cosines and Direction Ratios of a Line

#### 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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Solution Find the Direction Cosines of the Line Passing Through Two Points (−2, 4, −5) and (1, 2, 3) . Concept: Direction Cosines and Direction Ratios of a Line.
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