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Question

(C) The projection of a line 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 given by

Vedclass · Accepted Answer

$7$

Vedclass · Answer

(C) The projection of a line 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 given by the formula: $|l(x_2 - x_1) + m(y_2 - y_1) + n(z_2 - z_1)|$. Given points are $P(7, -5, 11)$ and $Q(-2, 8, 13)$. Direction cosines are $l = \frac{1}{3}, m = \frac{2}{3}, n = \frac{2}{3}$. Calculating the differences: $x_2 - x_1 = -2 - 7 = -9$ $y_2 - y_1 = 8 - (-5) = 13$ $z_2 - z_1 = 13 - 11 = 2$ Substituting these values into the formula: Projection $= |(\frac{1}{3})(-9) + (\frac{2}{3})(13) + (\frac{2}{3})(2)|$ $= |-\frac{9}{3} + \frac{26}{3} + \frac{4}{3}|$ $= |\frac{-9 + 26 + 4}{3}|$ $= |\frac{21}{3}| = 7$.

Find the projection of the line segment joining the points $P(7, -5, 11)$ and $Q(-2, 8, 13)$ on a line $AB$ having direction cosines $\frac{1}{3}, \frac{2}{3}, \frac{2}{3}$.

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