Question

Show that the line through the points (1, - 1, 2), (3, 4, - 2) is perpendicular to the line through the points (0, 3, 2) and (3, 5, 6).

Answer 1

We know that equation of a line passes through two points

$\frac{x - x_{1}}{x_{2} - x_{1}} = \frac{y - y_{1}}{y_{2} - y_{1}} = \frac{z - z_{1}}{z_{2} - z_{1}}$

Line from the points (1,-1,2) and (3,4,-2)

$\frac{x - 1}{2} = \frac{y + 1}{5} = \frac{z - 2}{ - 4}$
line through the points (0, 3, 2) and (3, 5, 6)

$\frac{x -0 }{3} = \frac{y - 3}{2} = \frac{z - 2}{4}$
If both the lines are perpendicular than

$a_{1}a_{2} + b_{1}b_{2} + c_{1}c_{2} = 0$
$a_{1} = 2 \\ \\ b_{1} = 5 \\ \\ c_{1} = - 4 \\ \\ a_{2} = 3 \\ \\ b_{2} = 2 \\ \\ c_{2} = 4 \\ \\ = 2(3) + 5(2) + ( - 4)(4) \\ \\ = 6 + 10 - 16 \\ \\ = 16 - 16 \\ \\ = 0 \\\\a_{1}a_{2} + b_{1}b_{2} + c_{1}c_{2} = 0$
Hence both the lines are perpendicular.