First slide

Planes in 3D

Question

The angle between the line $\bar{r} = (5 \bar{i} - \bar{j} - 4 \bar{k}) + λ (2 \bar{i} - \bar{j} + \bar{k})$ and the plane $\bar{r} . (3 \bar{i} - 4 \bar{j} - \bar{k}) + 5 = 0$ is

Moderate

A

$\sin^{- 1} (\frac{5}{2 \sqrt{91}})$
B

$\sin^{- 1} (\frac{9}{\sqrt{156}})$
C

$\sin^{- 1} (\frac{5}{\sqrt{156}})$
D

$\sin^{- 1} (\frac{9}{\sqrt{56}})$

Solution

From the line $(a_{1}, b_{1}, c_{1}) = z, - 1, 1$
From the plane $(a_{2}, b_{2}, c_{2}) = (3, - 4, - 1)$
$\sin θ = \frac{| a_{1} a_{2} + b_{1} b_{2} + c_{1} c_{2} |}{\sqrt{a_{1}^{2} + b_{1}^{2} + c_{1}^{2}} \sqrt{a_{2}^{2} + b_{2}^{2} + c_{2}^{2}}}$

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