First slide

Direction ratios and direction cosines

Question

A line makes angles $α, β, γ$ with positive axes, then the range of $\sum \sin α \sin β$ is

Moderate

A

$(- 1, 1]$
B

$(- 1, 2]$
C

$[- 1, 1)$
D

$(- 2, 1]$

Solution

If $α, β, γ$ are the angles made by a line ‘ $L$ ’ with the coordinate axes in the positive direction then $\cos^{2} α + \cos^{2} β + \cos^{2} γ = 1$ and $\sin^{2} α + \sin^{2} β + \sin^{2} γ = 2$
Consider, $\sum {(\sin α - \sin β)}^{2}$
It implies that
$\begin{matrix} \sum {(\sin α - \sin β)}^{2} \geq 0 \\ \sum \sin^{2} α + \sin^{2} β - 2 \sin α \sin β \geq 0 \\ 2 + 2 - 2 \sum \sin α \sin β \geq 0 \end{matrix}$
This gives $\sum \sin α \sin β \leq 2$

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