First slide

Direction ratios and direction cosines

Question

If $(1, - 1, 1)$ and $(2, - 3, 5)$ are direction ratios of two lines, then the angle between them is

Moderate

A

$\cos^{- 1} (\frac{5 \sqrt{2}}{\sqrt{57}})$
B

$\sin^{- 1} (\frac{5 \sqrt{2}}{\sqrt{57}})$
C

$\cos^{- 1} (\frac{50}{\sqrt{114}})$
D

$\tan^{- 1} (\frac{5 \sqrt{2}}{\sqrt{57}})$

Solution

If $θ$ is the acute angle between two lines having direction ratios $〈a_{1}, b_{1}, c_{1}〉$ and $〈a_{2}, b_{2}, c_{2}〉$
then $\cos θ = |\frac{a_{1} a_{2} + b_{1} b_{2} + c_{1} c_{2}}{\sqrt{{a_{1}}^{2} + {b_{1}}^{2} + {c_{1}}^{2}} \cdot \sqrt{{a_{2}}^{2} + {b_{2}}^{2} + {c_{2}}^{2}}}|$
Hence,
$\begin{matrix} \cos θ = \frac{2 + 3 + 5}{\sqrt{1 + 1 + 1} \cdot \sqrt{4 + 9 + 25}} \\ = \frac{10}{\sqrt{3} \cdot \sqrt{38}} \\ = \frac{5 \sqrt{2}}{\sqrt{57}} \end{matrix}$
Therefore, $θ = \cos^{- 1} (\frac{5 \sqrt{2}}{\sqrt{57}})$

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