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Sum

Show that no line in space can make angles `pi/6` and `pi/4` with X-axis and Y-axis.

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#### Solution

Let, if possible, a line in space make angles `pi/6 and pi/4` with X-axis and Y-axis.

∴ α = `pi/6, beta = pi/4`

Let the line make angle γ with Z-axis

∵ cos^{2}α + cos^{2}β + cos^{2}γ = 1

∴ `"cos"^2(pi/6) + "cos"^2(pi/4) + "cos"^2gamma = 1`

∴ `(sqrt3/2)^2 + (1/sqrt2)^2 + "cos"^2gamma = 1`

∴ `"cos"^2gamma = 1 - 3/4 - 1/2 = - 1/4`

This is not possible, because cos γ is real.

∴ cos^{2}γ cannot be negative.

Hence, there is no line in space which makes angles `pi/6 and pi/4` with X-axis and Y-axis.

Concept: Vectors and Their Types

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