If α, β, and γ are the angles with respect to OX, OY, and OZ, respectively, then their cosines satisfy the relationship
cos(α)² + cos(β)² + cos(γ)² = 1
Part A
For α = 45° and β = 60°, γ can be found from
cos(45°)² + cos(60°)² + cos(γ)² = 1
cos(γ)² = 1 - (1/√2)² - (1/2)² = 1/4
cos(γ) = ±1/2
γ = 60° or 120°
The inclination of V to OZ is 60° or 120°.
In rectangular coordinates,
V = (12cos(45°))i + (12cos(60°))j ± (12cos(60°))k
V = (6√2)i + 6j ± 6k
Part B
For V = 2i +3j +4k, the direction cosines and angles are
cos(α) = 2/||V|| = 2/√(2²+3²+4²) = 2/√29
α ≈ 68.2°
cos(β) = 3/√29
β ≈ 56.1°
cos(γ) = 4/√29
γ = 42.0°
