Question

Find the angle between the following pair of lines: `(-x+2)/(-2)=(y-1)/7=(z+3)/(-3)` and `(x+2)/(-1)=(2y-8)/4=(z-5)/4` and check whether the lines are parallel or perpendicular.

Answer 1

The given equations in standard from are
`(x-2)/(2)= (y-1)/(7)= (z+3)/(-3) " and " (x+2)/(-1) =(2y-8)/(y-4)/(2)=(z-5)/(4)`
Here `a_(1)= 2 ,b_(1)=7 , c_(1)= -3 " and " a_(2)=-1 ,b_(2) =2 ,c_(2) =4`
Let `theta ` be the angle between the given lines . Then
`" cos " theta = (|a_(1)a_(2)+b_(1)b_(2) +c_(1)c_(2)|)/((sqrt((a_(1)^(2)+b_(1)^(2)+c_(1)^(2))))(sqrt((a_(2)^(2)+b_(2)^(2)+c_(2)^(2)))))`
`=(|2xx(-1)+7xx2 +(-3)xx4|)/({sqrt(2^(2)+7^(2)+(-3)^(2)}}{sqrt((-1)^(2)+2^(2)+4^(2))}}`
`=(|(-2)+14+(-12)|)/((sqrt(62))(sqrt(21)))=0`
`:. theta =(pi)/(2)`
Hence the angle between the given lines is `(pi)/(2)`