Find the angle between the following pair of lines: (-x + 2)/-2 = (y-1)/7 = (z + 3)/-3

Question

Find the angle between the following pair of lines:

\(\frac{-x+2}{-2}=\frac{y-1}{7}=\frac{z+3}{-3}\) and \(\frac{x+2}{-1}=\frac{2y-8}{4}=\frac{z-5}{4}\)

and check whether the lines are parallel or perpendicular.

Answer 1

The equation of given lines can be written in standard form as

If \(\vec{b_1}\) and \(\vec{b_2}\) are vectors parallel to lines (i) and (ii) respectively, then

\(\vec{b_1}=2\hat{i}+7\hat{j}-3\hat{k}\) and \(\vec{b_2}=-\hat{i}+2\hat{j}+4\hat{k}\)

Obviously, if q is the angle between lines (i) and (ii) then q is also the angle between \(\vec{b_1}\) and \(\vec{b_2}.\)

Angle between both lines is \(90^\circ.\)

Hence, given lines are perpendicular to each other.