If a vector P making an angle α, β and γ respectively with the . . .

Question : If a vector P making an angle α, β and γ respectively with the x, y and z axes respectively then sin²α + sin²β + sin²γ = 

Doubt by Muskan

Solution :

When we resolve the vector A in Three dimensions such that vector A makes angle α, β and γ respectively with the x, y and z axes respectively then

A_x = Acosα — (1)
P_y = Acosβ — (2)
A_z = Acosγ — (3) 

Squaring and adding the above equations

A_x²+A_y²+A_z² = A²cos²α+A²cos²β+A²cos²γ

A² = A²[cos2α+cos2β+cos2γ]
[∵A=√(A_x²+A_y²+A_z²)
So A²=A_x²+A_y²+A_z²]
1 = cos²α+cos²β+cos²γ

cos²α+cos²β+cos²γ = 1

Here cosα, cosβ and cosγ are called the direction cosines of vector P.

Now,

We know,
cos²α+cos²β+cos²γ = 1
1-sin²α + 1- sin²β + 1-sin²γ =1
[∵ cos²θ=1-sin²θ]
3-1= sin²α + sin²β + sin²γ
2 = sin²α + sin²β + sin²γ

Hence, sin²α + sin²β + sin²γ = 2