First slide

Direction ratios and direction cosines

Question

A line makes angles $α, β, γ, δ$ with the diagonals of a cube. Then $\sum \cos^{2} α = \frac{1}{λ}$ where $λ$ =

Very Easy

A

$\frac{3}{2}$
B

$\frac{3}{4}$
C

$\frac{3}{8}$
D

None

Solution

$\begin{array}{l} let l be the line having directions cosines 〈l,m,n〉 \\ diagonals of cube are OP,CS,BQ,AR \\ direction ratios OP are 〈1,1,1〉 \\ direction ratios CS are 〈1,1,-1〉 \\ direction ratios BQ are 〈1,-1,1〉 \\ direction ratios AR are 〈-1,1,1〉 \\ Given α,β,γ,δ are angles made by the line with diagonals \\ cosα= \frac{l (1) +m (1) +n (1)}{\sqrt{l^{2} {+m}^{2} {+n}^{2}} \sqrt{1^{2} {+1}^{2} {+1}^{2}}} \\ = \frac{l+m+n}{\sqrt{3}} \\ cosβ= \frac{l+m+n}{\sqrt{3}},cosγ= \frac{l-m+n}{\sqrt{3}},cosδ= \frac{-l+m+n}{\sqrt{3}} \\ \cos^{2} α + \cos^{2} β + \cos^{2} γ + \cos^{2} δ \\ = \frac{{(l + m + n)}^{2}}{3} + \frac{{(l + m - n)}^{2}}{3} + \frac{{(l - m + n)}^{2}}{3} + \frac{{(- l + m + n)}^{2}}{3} \\ = \frac{1}{3} + \frac{1}{3} + \frac{1}{3} + \frac{1}{3} \\ = \frac{4}{3} \end{array}$

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