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Three Dimensional Geometry

Question
CBSEENMA12033398
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Find the vector equation of the straight line passing through (1, 2, 3) and perpendicular to the plane straight r with rightwards arrow on top. space open parentheses straight i with hat on top space plus space 2 space straight j with hat on top space minus space 5 space straight k with hat on top close parentheses space plus space 9 space equals 0 space.

Solution
The required line passes through (1, 2, 3)
therefore space space space space space space space space straight a with rightwards arrow on top space equals space straight i with hat on top space plus space 2 space straight j with hat on top space plus space 3 space straight k with hat on top
Since required line is perpendicular to the plane
straight r with rightwards arrow on top space equals space left parenthesis straight i with hat on top space plus space 2 space straight j with hat on top space minus space 5 space straight k with hat on top right parenthesis space plus space 9 space equals 0
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      therefore space space equation space of space line space is
                       straight r with rightwards arrow on top space equals space straight a with rightwards arrow on top space plus space straight lambda straight b with rightwards arrow on top
or                    straight r with rightwards arrow on top space equals space open parentheses straight i with hat on top space plus space 2 space straight j with hat on top space plus space space 3 space straight k with hat on top close parentheses space plus space straight lambda space left parenthesis straight i with hat on top space plus space 2 space straight j with hat on top space minus space 5 space straight k with hat on top right parenthesis

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If α, β, γ are the angles which a line makes with the axes, prove that sin2α + sin2 β + sin2γ = 2.

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