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

Given, the line makes the angles α, β and γ respectively with x-axis, y-axis and z-axis.

As per the relation between direction cosines of a line, l² + m² + n² = 1 where l, m, n are the direction cosines of a line from x-axis, y-axis and z-axis respectively.

So, we can say that,

cos²α + cos²β + cos²γ = 1 ------ (1)

Now, we should find the value for

cos2α + cos2β + cos2γ

cos2α can be written as 2cos²α -1,

cos2α + cos2β + cos2γ

= (2cos²α -1) + (2cos²β -1) + (2cos 2γ -1)

= 2 (cos²α+ cos²β + cos²γ) – 3

= 2(1) – 3

[From Equation (1)]

= -1

Therefore,

cos2α + cos2β + cos2γ = -1