Question

If a vector makes angles α, β, γ with OX, OY and OZ respectively, then write the value of sin² α + sin² β + sin² γ.

Answer 1

Suppose, a vector \[\overrightarrow{OP}\] makes an angle \[\alpha, \beta, \gamma\] with OX, OY, OZ respectively.
Then, direction cosines of the vector are given by \[l = \cos \alpha , m = \cos \beta \text{ and }n = \cos \gamma\]
Consider, 
\[\sin^2 \alpha + \sin^2 \beta + \sin^2 \gamma = 1 - \cos^2 \alpha + 1 - \cos^2 \beta + 1 - \cos^2 \gamma\]
\[ = 3 - \left( \cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma \right)\]
\[ = 3 - \left( l^2 + m^2 + n^2 \right)\]
= 3 - 1                                [∵ \[l^2 + m^2 + n^2 = 1\]]

= 2