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

Toolbox:
• Direction cosines of vector $x\hat i+y\hat j+z\hat k$ are $l=\large\frac{x}{\sqrt{x^2+y^2+z^2}}$$,m=\large\frac{y}{\sqrt{x^2+y^2+z^2}}$$,n=\large\frac{z}{\sqrt{x^2+y^2+z^2}}$
Step 1:
Let $P(-2,4,-5)$ and $Q(1,2,3)$
$\overrightarrow{ PQ}=\overrightarrow{OQ}-\overrightarrow{OP}$
$\quad\;\;=(\hat i+2\hat j+3\hat k)-(-2\hat i+4\hat j-5\hat k)$
$\quad\;\;=3\hat i-2\hat j+8\hat k$
$\mid \overrightarrow {PQ}\mid=\sqrt{(3)^2+(-2)^2+(8)^2}$
$\qquad\;=\sqrt{9+4+64}$
$\qquad\;=\sqrt{77}$
Step 2:
Direction cosines of the given vector is
$l=\large\frac{x}{\sqrt{x^2+y^2+z^2}}=\frac{3}{\sqrt{77}}$
$m=\large\frac{y}{\sqrt{x^2+y^2+z^2}}=\frac{-2}{\sqrt{77}}$
$n=\large\frac{z}{\sqrt{x^2+y^2+z^2}}=\frac{8}{\sqrt{77}}$