First slide

Direction ratios and direction cosines

Question

The angle between a line with direction ratios 2:2:1 and a line joining (3,1,4) to (7,2,12) is

Moderate

A

$\cos^{- 1} (2 / 3)$
B

$\cos^{- 1} (- 2 / 3)$
C

$\tan^{- 1} (- 2 / 3)$
D

$\sin^{- 1} (2 / 3)$

Solution

D.R's = (2,2,1), and (7-3, 2-12, 12-4) = (2,2,1), (4,1,8)
$\cos θ = \frac{a_{1} a_{2} + b_{1} b_{2} + c_{1} c_{2}}{\sqrt{a_{1}^{2} + b_{1}^{2} + c_{1}^{2}} \sqrt{a_{2}^{2} + b_{2}^{2} + c_{2}^{2}}} = \frac{2}{3}$
$∴ θ = \cos^{- 1} (\frac{2}{3})$

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