The acute angle between the line joining the points $(2,1,-3),(-3,1,7)$ and a line parallel to…
The acute angle between the line joining the points $(2,1,-3),(-3,1,7)$ and a line parallel to $\frac{x-1}{3}=\frac{y}{4}=\frac{z+3}{5}$ through the point $(-1,0,4)$ is
$\cos ^{-1}\left(\frac{1}{\sqrt{10}}\right)$
$\cos ^{-1}\left(\frac{5}{7 \sqrt{10}}\right)$
$\cos ^{-1}\left(\frac{7}{5 \sqrt{10}}\right)$
$\cos ^{-1}\left(\frac{3}{5 \sqrt{10}}\right)$
Solution
The d.r.s. of the line joining the points $(2,1,-3)$ and $(-3,1,7)$ are $-5,0,10$
The d.r.s. of the line parallel to line $\frac{x-1}{3}=\frac{y}{4}=\frac{z+3}{5}$ are $3,4,5$
$\therefore \quad$ The angle between the lines having d.r.s.
$-5,0,10$ and $3,4,5$ is
$\cos \theta=\left|\frac{-5(3)+0(4)+10(5)}{\sqrt{25+0+100} \sqrt{9+16+25}}\right|$
$\begin{aligned} & \Rightarrow \cos \theta=\frac{35}{25 \sqrt{10}} \\ & \Rightarrow \theta=\cos ^{-1}\left(\frac{7}{5 \sqrt{10}}\right)\end{aligned}$