Problem 19 - Entrance Test

The points (4, 7, 8), (2, 3, 4) and (-1, -2, 1) are:

A. CollinearB. Vertices of a right angled triangleC. Vertices of an equilateral triangleD. None of these

Correct: A

Let A = (4, 7, 8), B = (2, 3, 4), and C = (-1, -2, 1). Vector AB = B - A = (2-4, 3-7, 4-8) = (-2, -4, -4). Vector BC = C - B = (-1-2, -2-3, 1-4) = (-3, -5, -3). Vector AC = C-A = (-1-4, -2-7, 1-8) = (-5,-9,-7). If they are collinear AB = kBC, -2/-3 = -4/-5, this is not true. AB + BC does not = Ac and AB = kAC; -2=-5k, -4 = -9k -> so not collinear. Distance between A and B = sqrt(4+16+16) = sqrt(36) = 6. Distance between B and C = sqrt(9+25+9) = sqrt(43). Distance between A and C = sqrt ( 25+81+49) = sqrt(155). 6^2 +sqrt(43) is NOT = sqrt(155), so it's not right angle or equilateral Triangle. They are collinear. The direction vectors AB = -2,-4, -4 -> -1,-2,-2, and AC is: -5,-9,-7. If AB, BC are collinear then BC = k * AB. -3=-2K -> k = 3/2, -5 = 3/2 4 -> so no equal to BC are not linear either