Question

If the points a b c are (2,1,-1),(4,-2,3),and (-2,6,3) respectively.Then find the area of triangle abc

Answer 1

Vector AB is the vector that takes you from A to B. Therefore, A + AB = B (A and B are position vectors), so AB = B - A = (4,3,-2)-(2,-1,1) = (4-2,3+1,-2-1) = (2,4,-3). In the same way, AC = (3,0,-3)-(2,-1,1) = (3-2,1,-3-1) = (1,1,-4).

AB x AC is defined as the determinant of the matrix [i,j,k; AB_1,AB_2,AB_3; AC_1,AC_2,AC_3], so we get (4*(-4)-(-3)1, -(2(-4)-(-3)1), 21-4*1) = (-16+3,-(-8+3),2-4) = (-13, 5, -2).

The magnitude of AB x AC is equal to |AB||AC|sin(theta), where theta is the angle between AB and AC. With the help of a diagram we can show that |AC|sin(theta) is the height of triangle ABC, and therefore its area is .5 * base * height = .5|AB||AC|sin(theta) = .5|AB x AC| = 1/2sqrt((-13)^2+5^2+(-2)^2) = 1/2sqrt(169+25+4) = 1/2sqrt(198).