Does the point D(-3, 4, -6)lie in the same plane as the triangle with vertices A(0, 0, 0), B(5, 5, 35), C(5, 4, 33)?
A)No
B)Yes

Question

Quizplus · Accepted Answer

To determine if point D lies in the same plane as the triangle formed by points A, B, and C, we can use the concept of vectors and the scalar triple product. The scalar triple product of vectors AB, AC, and AD gives the volume of the parallelepiped formed by these vectors. If the volume is zero, it means that all four points lie in the same plane.Vectors AB, AC, and AD can be found as follows:- AB = B - A = (5, 5, 35)- AC = C - A = (5, 4, 33)- AD = D - A = (-3, 4, -6)The scalar triple product is given by the determinant of the matrix formed by these vectors:|  5  5  35 ||  5  4  33 || -3  4  -6 |Calculating the determinant:= 5(4*(-6) - 33*4) - 5(5*(-6) - 33*(-3)) + 35(5*4 - 4*5)= 5(-24 - 132) - 5(-30 + 99) + 35(20 - 20)= 5(-156) - 5(69) + 35(0)= -780 - 345= -1125Since the scalar triple product is not zero (-1125 ≠ 0), the volume of the parallelepiped is not zero, which means that point D does not lie in the same plane as the triangle formed by points A, B, and C.

Does the Point D(-3, 4, -6)lie in the Same Plane