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Question
If α, β, γ are direction angles of a line and α = 60°, β = 45°, then γ = ______.
Options
30° or 90°
45° or 60°
90° or 130°
60° or 120°
Solution
If α, β, γ are direction angles of a line and α = 60°, β = 45°, γ = 60° or 120°.
Explanation:
We have
cos2α + cos2β + cos2γ = 1
⇒ cos260° + cos245° + cos2γ = 1
⇒ `(1/2)^2 + (1/sqrt2)^2 + cos^2 gamma = 1`
⇒ `1/4 + 1/2 + cos^2 gamma = 1`
⇒ `cos^2 gamma = 1 - 3/4 = 1/4`
⇒ `cos gamma = +- 1/2`
⇒ γ = 60° or 120°
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