If a line makes angles 60°, 60°, and γ with the x, y,

Solution

Let the direction cosines of the line be l,m,n. The angles made by the line with the x, y, z axes are α,β,γ respectively. We have the relations: l=cosα m=cosβ n=cosγ Given that the line makes angles 60°,60°, and γ with the x, y, z axes respectively. So, α=60°, β=60°. The direction cosines are: l =cos60°= 1/2 m=cos60°= 1/2 n=cosγ We know that the sum of the squares of the direction cosines is equal to 1: l²+m²+n²=1 Substitute the values of l and m: (1/2​)²+(1/2​)²+(cosγ)²=1 1/4 + 1/4​+cos²γ=1 2/4​+cos²γ=1 1/2​+cos²γ=1 cos²γ=1− 1/2​ cos²γ= 1/2​ Taking the square root of both sides: cosγ = ±1/√2 Therefore, γ = 45° or 135° But direction angles are taken in the range 0° to 180° , and the question seeks the angle γ between the line and the z-axis, which is the acute angle between them .