Avector v is inclined to an angle ox at 45degree and oy at 60degreee find its inclination to oz, find its magnitude if v is 12units express v in terms of ai+bj+ck. note v is given as 2i+3j+4k. the direction ratio is 2: 3: 4

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°

Volume = the base times the height. i’m not going your work for you, but this is all you need to know.