Question

If a vector has direction angles 45° and 60°, find the third direction angle.

Answer 1

Let α, β, γ be the angles made by the vector with positive directions of X, Y, Z axes respectively.

∴ α = 45°, β = 60°

We know that,

∵ cos² α + cos² β + cos² γ = 1

∴ cos²45° + cos²60° + cos²r = 1

∴ `(1/sqrt2)^2 + (1/2)^2 + "cos"^2gamma = 1` 

∴ `1/2 + 1/4 + cos^2gamma` = 1

∴ `"cos"^2gamma = 1 - 1/2 - 1/4`

∴ `"cos"^2gamma = 1/4`

∴ cos γ = `+- 1/2`

∴ cos γ = `1/2`  or  cos γ = `- 1/2`

∴ cos γ = `pi/3` or cos γ = − `pi/3`

∴ cos`(pi - pi/3) = "cos"(2pi)/3` 

∴ `gamma = pi/3` or `gamma = (2pi)/3`

Hence, the third direction angle is `pi/3` or `(2pi)/3`