Question

If a line makes angles α, β, γ, δ with four diagonals of a cube, then cos² α + cos² β + cos²γ + cos² δ is equal to

Options

• \[\frac{1}{3}\]

• \[\frac{2}{3}\]

• \[\frac{4}{3}\]

• \[\frac{8}{3}\]

Answer 1

\[\frac{4}{3}\] 

\[\text { Let a be the length of an edge of the cube and let one corner be at the origin as shown in the figure . Clearly, OP, AR, BS and CQ are the diagonals of the cube } . \]

\[\text{ The direction ratios of OP, AR, BS and CQ are } \]

\[a - 0, a - 0, a - 0, \text{ i . e } . a, a, a\]

\[0 - a, a - 0, a - 0, \text{ i . e } . - a, a, a\]

\[a - 0, 0 - a, a - 0,\text{ i . e } . a, - a, a\]

\[a - 0, a - 0, 0 - a\text{ i . e } . a, a, - a \]

\[ \text { Let the direction ratios of a line be proportional to l, m and n . Suppose this line makes angles} \alpha, \beta, \gamma \text { and } \delta \text{ with OP, AR, BS and CQ, respectively  i . e } . \]

\[\text{ Now} , \alpha \text{ is the angle between OP and the line whose direction ratios are proportional to l, m and n } . \]

\[ \cos \alpha = \frac{a . l + a . m + a . n}{\sqrt{a^2 + a^2 + a^2}\sqrt{l^2 + m^2 + n^2}} \Rightarrow \cos \alpha = \frac{l + m + n}{\sqrt{3}\sqrt{l^2 + m^2 + n^2}}\]

\[\text{ Since } \beta \text{ is the angle between AR and the line with direction ratios proportional to l, m and n, we get }\]

\[ \cos \beta = \frac{- a . l + a . m + a . n}{\sqrt{a^2 + a^2 + a^2}\sqrt{l^2 + m^2 + n^2}} \Rightarrow \cos \beta = \frac{- l + m + n}{\sqrt{3}\sqrt{l^2 + m^2 + n^2}}\]

\[\text{ Similarly }, \]

\[ \cos \gamma = \frac{a . l - a . m + a . n}{\sqrt{a^2 + a^2 + a^2}\sqrt{l^2 + m^2 + n^2}} \Rightarrow \cos \gamma = \frac{l - m + n}{\sqrt{3}\sqrt{l^2 + m^2 + n^2}}\] 

\[ \cos \delta = \frac{a . l + a . m - a . n}{\sqrt{a^2 + a^2 + a^2}\sqrt{l^2 + m^2 + n^2}} \Rightarrow \cos \delta = \frac{l + m - n}{\sqrt{3}\sqrt{l^2 + m^2 + n^2}}\]

 

\[ \cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma + \cos^2 \delta \]

\[ = \frac{\left( l + m + n \right)^2}{3\left( l^2 + m^2 + n^2 \right)} + \frac{\left( - l + m + n \right)^2}{3\left( l^2 + m^2 + n^2 \right)} + \frac{\left( I - m + n \right)^2}{3\left( l^2 + m^2 + n^2 \right)} + \frac{\left( l + m - n \right)^2}{\sqrt{3}\sqrt{l^2 + m^2 + n^2}}\]

\[ = \frac{1}{3\left( l^2 + m^2 + n^2 \right)}\left\{ \left( l + m + n \right)^2 + \left( - l + m + n \right)^2 + \left( I - m + n \right)^2 + \left( l + m - n \right)^2 \right\}\]

\[ = \frac{1}{3\left( l^2 + m^2 + n^2 \right)}4\left( l^2 + m^2 + n^2 \right) = \frac{4}{3}\]