First slide

Direction ratios and direction cosines

Question

The projection of a segment joining points $P (x_{1}, y_{1}, z_{1})$ and $Q (x_{2}, y_{2}, z_{2})$ on a line with direction cosines $l, m, n$ is $a l + b m + c n$ . Then $a^{2} + b^{2} + c^{2}$ =

Moderate

A

$P Q^{2}$
B

$P Q$
C

1
D

None

Solution

The projection of a segment joining points $P (x_{1}, y_{1}, z_{1})$ and $Q (x_{2}, y_{2}, z_{2})$ on a line with direction cosines $l, m, n$ is $l (x_{2} - x_{1}) + m (y_{2} - y_{1}) + n (z_{1} - z_{2})$
Hence,
$a^{2} + b^{2} + c^{2} = {(x_{2} - x_{1})}^{2} + {(y_{2} - y_{1})}^{2} + {(z_{2} - z_{1})}^{2} = {(P Q)}^{2}$

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