If a vector has direction angles α = π/4 and β = π/3, find the third direction angle γ.

Solution:

Consider the position vector \(OP\vec{} or (r\vec{})\) of a point P(x, y, z) as in the diagram above. The angles ɑ, β, and γ made by the vector \(r\vec{}\) with the positive directions of x, y, and z axes respectively are called its direction angles. The cosine values of these angles i.e. Cos(ɑ), Cos(β), and Cos(γ) are called the directions cosines of the vector \(r\vec{}\) , usually denoted by l, m, and n respectively. The fundamental equation can be therefore be also written as:

l² + m² + n² = 1 (where l =Cos(ɑ), m = Cos(β), n = Cos(γ) )

This problem is related to the fundamental relation known as the direction cosines of Vector \(r\vec{}\). The relationship is stated as follows:

cos²ɑ + cos²β + cos²γ = 1 --- (1)

Where ɑ, β, and γ are the angles made by the vector \(r\vec{}\) with the positive directions of x, y, and z axes respectively.

Two of the angles are given as ɑ = π/4 and β = π/3.

cos²π/4 + cos²π/3. + cos²γ = 1 --- (2)

cos(π/4) = 1/√2 and cos(π/3) = 1/2

Substituting the values in equation (2) we have

(1/√2)² + (1/2)² + cos²γ = 1

1/2 + 1/4 + cos²γ = 1

3/4 + cos²γ = 1

cos²γ = 1 - 3/4 = 1/4

cosγ = √1/4 = ± 1/2

⇒ γ = π/3

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If a vector has direction angles α = π/4 and β = π/3, find the third direction angle γ.

Summary:

If a vector has direction angles α = π/4 and β = π/3, then the third direction angle γ = π/3.