Direction ratios and direction cosines

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Question

$P (1, 2, - 2,)$ , $Q (8, 10, 11)$ , $R (1, 2, 3)$ , $S (3, 5, 7)$ if $λ$ denotes the length of projection of PQ on RS then $29 λ^{2} + 29$ is equal to

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Solution

D.c's of $R S = (\frac{2}{\sqrt{29}}, \frac{3}{\sqrt{29}}, \frac{4}{\sqrt{29}}) = (l, m, n)$
$λ = | l (x_{1} - x_{2}) + m (y_{1} - y_{2}) + n (z_{1} - z_{2}) |$ where P,Q is $(x_{1}, y_{1}, z_{1}) (x_{2}, y_{2}, z_{2})$

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