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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 .
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