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Question
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`
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