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Show that 2 Lines are Perpendicular:

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Show that the line L1 through A(3, -2, 4) and B(5, 1, -2) is pperpendicular to L2 through C(-2, 3, 1) and D(-5, 1, -1).

If the lines are perpendicular, the dot product of the vectors describing them is 0.

For L1, the two points give the vector AB = [5 - 3]i + [1 - (-2)]j + [-2 - 4]k = 2i + 3j - 6k

For L2, the two points give CD = [-5 - (-2)]i + [1 - 3]j + [-1 - 1]k = -3i - 2j - 2k.

Now calculate the dot product:

AB · CD = [2 * (-3)] + [3 * (-2)] + [(-6) * (-2)] = 0

Hence the lines are perpendicular.