First slide

Direction ratios and direction cosines

Question

If $P$ and $Q$ are complementary angles and $P, Q, R$ are angles made by a straight line with $X -, Y -$ and $Z -$ axes respectively, then $R =$

Moderate

A

$π$
B

$\frac{π}{2}$
C

$\frac{π}{4}$
D

$\frac{3 π}{4}$

Solution

If $α, β, γ$ are the angles made by a line ‘L’ with the coordinate axes in the positive direction then
Given $α = P, β = \frac{π}{2} - P, γ = R$
Hence,
$\begin{matrix} \cos^{2} P + \cos^{2} (\frac{π}{2} - P) + \cos^{2} R = 1 \\ \cos^{2} P + \sin^{2} P + \cos^{2} R = 1 \\ \cos^{2} R = 0 \end{matrix}$
Therefore, $R = \frac{π}{2}$

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