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

The given points are A(1, -1, 2), B(3, 4, - 2), C(0, 3,2) and D(3, 5, 6) direction ratios of AB are
(3 - 1, 4 - (-1), - 2 - 2) = (2,4 + 1, -4) = (2, 5, - 4)
and direction ratios of CD are (3 - 0, 5 - 3, 6 - 2) or (3, 2, 4).
We know that two lines AB and CD with direction ratios, $a_1,b_1,c_{1}and{a}_2,b_2,c_2$ are perpendicular if $a_1{a}_2+b_1{b}_2+c_1{c}_2=0$
$\therefore 2\times 3+5\times 2+(-4)\times 4=6+10-16=0$
Therefore, the lines AB and CD are at right angles.