Question

If a line makes angles α, β and γ with the coordinate axes, find the value of cos 2α + cos 2β + cos 2γ.

Answer 1

$$\begin{gathered} \mathrm{It}\mathrm{is}\mathrm{given}\mathrm{that}\mathrm{the}\mathrm{the}\mathrm{line}\mathrm{makes}\mathrm{angles}\mathrm{\alpha },\mathrm{\beta },\mathrm{\gamma }\mathrm{with}\mathrm{the}\mathrm{coordinate}\mathrm{axis}. \\ \therefore l=\mathit{cos}\mathit{}\alpha \mathit{,}\mathit{}m=\mathit{cos}\mathit{}\beta \mathit{}\mathrm{and}\mathit{}n=\mathit{cos}\mathit{}\gamma \\ \Rightarrow l^{\mathit{2}}+m^{\mathit{2}}+n^{\mathit{2}}=1 \\ \Rightarrow {\cos }^2\mathrm{\alpha }+{\cos }^2\mathrm{\beta }+{\cos }^2\mathrm{\gamma }=1...\left(1\right) \\ \mathrm{Now}, \\ \cos 2\mathrm{\alpha }+\cos 2\mathrm{\beta }+\cos 2\mathrm{\gamma }=\left(2{\cos }^2\mathrm{\alpha }-1\right)+\left(2{\cos }^2\mathrm{\beta }-1\right)+\left(2{\cos }^2\mathrm{\gamma }-1\right) \\ =2\left({\cos }^2\mathrm{\alpha }+{\cos }^2\mathrm{\beta }+{\cos }^2\mathrm{\gamma }\right)-3 \\ =2\left(1\right)-3\left[\mathrm{From}\left(1\right)\right] \\ =-1 \end{gathered}$$