posted by samantha .
The person who helped me before was not very helpful.
ABC has vertices A(3,2,-5), B(4,-1,7) and C(-8,3,-6).
a) Determine the area of ABC.
b) Determine the coordinates of point D such that ABCD is a parallelogram.
c) Is the parallelogram a rectangle? Justify your answer.
Given two vectors u and v, forming two sides of a parallelogram, u+v is the corner opposite (0,0). Here, we can take
u = B-A = (1,-3,12)
v = C-A = (11,1,-1)
The area of ∆ (not ) ABC is half the area of the parallelogram ABCD. So, it is
1/2 |uxv| = 1/2 |(-9,133,34)| = 1/2 √18926 = 68.79
The parallelogram is not a rectangle since
u•v = 11-3-12 = -4 ≠ 0