Maharashtra State BoardHSC Arts 12th Board Exam
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If a vector has direction angles 45° and 60°, find the third direction angle. - Mathematics and Statistics

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Sum

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

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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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