the vector u makes angles α(alpha), β(beta), γ(gama) with respectively the X–axis, the Y –axis and the Z–axis. Find γ if α = 2π/3 and β = π /4

This question deals with direction cosines

you must have come across the property
cos^2 α + cos^2 β + cos^2 γ = 1
cos^2 (2π/3) + cos^2 (π/4) + cos^2 γ = 1
1/4 + 1/2 + cos^2 γ = 1
cos^2 γ = 1 - 1/2 - 1/4
cos^2 γ = 1/4
cos γ = 1/2
γ = π/6 using only the first quadrant case

To find γ, we can first find the angle between the vector u and the Z-axis.

Let's use the fact that the sum of the angles between a vector and the coordinate axes is always equal to 90 degrees (or π/2 radians).

So, the angle between the vector u and the Z-axis is given by:
γ = π/2 - (α + β) = π/2 - (2π/3 + π/4)

To simplify, let's find a common denominator for 3 and 4, which is 12:
γ = π/2 - (8π/12 + 3π/12)
= π/2 - (11π/12)

To subtract the fractions, we need a common denominator of 12:
γ = (6π/12) - (11π/12)
= -5π/12

Therefore, the angle γ is -5π/12.