First slide

Direction ratios and direction cosines

Question

If a line makes acute angles $α, β, γ$ with the coordinate axes such that $\cos α \cos β = \frac{1}{\sqrt{6}}; \cos β \cos γ = \frac{1}{2 \sqrt{3}}; \cos γ \cos α = \frac{1}{3 \sqrt{2}}$ then , $3 {(\cos α + \cos β + \cos γ)}^{2} = \sqrt{2} + \sqrt{3} + \sqrt{6} + \sqrt{λ}$ then $λ =$

Difficult

A

6
B

7
C

8
D

9

Solution

If $α, β, γ$ are the angles made by a line with axes then $\cos^{2} α + \cos^{2} β + \cos^{2} γ = 1$
Consider,
$3 {(\cos α + \cos β + \cos γ)}^{2}$
$\begin{matrix} = 3 (\cos^{2} α + \cos^{2} β + \cos^{2} γ + 2 \cos α \cos β + 2 \cos β \cos γ + 2 \cos γ \cos α) \\ = 3 (1 + \frac{2}{\sqrt{6}} + \frac{2}{2 \sqrt{3}} + \frac{2}{3 \sqrt{2}}) \\ = 3 + \sqrt{6} + \sqrt{3} + \sqrt{2} \end{matrix}$
Therefore, $\sqrt{λ} = 3 \Rightarrow λ = 9$

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