If cos α, cos β, cos γ are the direction cosines of vector $\vec{a}$, then the value of cos 2α + cos 2β + cos 2γ is equal to :

-1

cos α, cos β, cos γ → direction ratios of $\vec{a}$ so

cos²α + cos²β + cos²γ = 1        ........(1)

so cos 2α + cos 2β + cos 2γ

as cos 2x = 2cos²x - 1

= 2cos²α - 1 + cos²β - 1 + 2cos²γ - 1

= 2(cos²α + cos²β + cos²γ) - 3

= 2 × 1  - 3

= -1