Find the direction cosines of the line passin | vedclass

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(A) The direction cosines of a line passing through two points $P(x_1, y_1, z_1)$ and $Q(x_2, y_2, z_2)$ are given by $\frac{x_2-x_1}{PQ}, \frac{y_2-y

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$\frac{3}{\sqrt{77}}, \frac{-2}{\sqrt{77}}, \frac{8}{\sqrt{77}}$

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(A) The direction cosines of a line passing through two points $P(x_1, y_1, z_1)$ and $Q(x_2, y_2, z_2)$ are given by $\frac{x_2-x_1}{PQ}, \frac{y_2-y_1}{PQ}, \frac{z_2-z_1}{PQ}$,where $PQ = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2}$.Given points are $P(-2, 4, -5)$ and $Q(1, 2, 3)$.First,calculate the distance $PQ$:$PQ = \sqrt{(1 - (-2))^2 + (2 - 4)^2 + (3 - (-5))^2} = \sqrt{(3)^2 + (-2)^2 + (8)^2} = \sqrt{9 + 4 + 64} = \sqrt{77}$.Now,the direction cosines are:$l = \frac{1 - (-2)}{\sqrt{77}} = \frac{3}{\sqrt{77}}$$m = \frac{2 - 4}{\sqrt{77}} = \frac{-2}{\sqrt{77}}$$n = \frac{3 - (-5)}{\sqrt{77}} = \frac{8}{\sqrt{77}}$Thus,the direction cosines are $\frac{3}{\sqrt{77}}, \frac{-2}{\sqrt{77}}, \frac{8}{\sqrt{77}}$.

Find the direction cosines of the line passing through the two points $(-2, 4, -5)$ and $(1, 2, 3)$.

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