Question

Find the direction cosines and direction ratios of the line joining the points
$A(5,6,-3),B(1,-6,3)$

Answer 1

Given points are $A=\left(5,6,-3\right)\&B=\left(1,-6,3\right)$

Direction ratios are $a=x_2-x_1=1-5=-4,b=y_2-y_1=-6-6=-12,c=z_2-z_1=3+3=6,$

Now, $\sqrt{a^2+b^2+c^2}=\sqrt{{\left(-4\right)}^2+{\left(-12\right)}^2+6^2}=\sqrt{16+144+36}=14$

Direction cosines are

$l=\frac{a}{\sqrt{a^2+b^2+c^2}}=\frac{-4}{14}=-\frac{2}{7}m=\frac{b}{\sqrt{a^2+b^2+c^2}}=\frac{-12}{14}=-\frac{6}{7}n=\frac{c}{\sqrt{a^2+b^2+c^2}}=\frac{6}{14}=\frac{3}{7}$

Direction ratios are $\left(-4,-12,6\right)$ and direction cosines are $\left(-\frac{2}{7},\frac{-6}{7},\frac{3}{7}\right)$