Question

If $A(x_1,y_1,z_1)$ and $B(x_2,y_2,z_2)$ are two points such that the direction cosines of $\overline{AB}$ are $l$, $m$, $n$ then
$l=\frac{x_2-x_1}{\left|\overline{AB}\right|}$, $m=\frac{y_2-y_1}{\left|\overline{AB}\right|}$, $n=\frac{z_2-z_1}{\left|\overline{AB}\right|}$.

Answer 1

Direction cosines of $\overline{AB}$ is the cosines of angles made by the line with the positive direction of $x$- axis, $y$-axis and $z$- axis respectively.

Let, $\alpha$, $\beta$, $\gamma$ be the angles made by the line with the positive direction of $x$-axis, $y$-axis and $z$-axis.

Then $l=\cos \alpha =\frac{x_2-x_1}{\sqrt{(x_2-x_1{)}^2+(y_2-y_1{)}^2+(z_2-z_1{)}^2}}=\frac{(x_2-x_1)}{|\overline{AB}|}.$

and $m=\cos \beta =\frac{y_2-y_1}{\sqrt{(x_2-x_1{)}^2+(y_2-y_1{)}^2+(z_2-z_1{)}^2}}=\frac{(y_2-y_1)}{|\overline{AB}|}.$

and $n=\cos \alpha =\frac{z_2-z_1}{\sqrt{(x_2-x_1{)}^2+(y_2-y_1{)}^2+(z_2-z_1{)}^2}}=\frac{(z_2-z_1)}{|\overline{AB}|}.$