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

Sum

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

#### Solution

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

Concept: Scalar Triple Product of Vectors
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