Vector Calculus help. Let A=(1,2,-1), B=(-1,1,1) and C=(-3,2,1)?

Question

Find the area of triangle ABC.

Find the equation of the plane through A, B, and C.

Find the equation of the plane through the point P(1,1,1) and parallel to the plane of the triangle ABC

Answer 1

1.)find the length of AB,BC,Ca. using heros method find the area.
2) we have eqn of plane as ax+by+cz =0
put values of A,B,C and solve the three eqn.
x+2y+z =0
-x+y+z =0
-3x+2y +z =0

Answer 2

Given the points A(1,2,-1); B(-1,1,1); and C=(-3,2,1), find the area of triangle ABC.

First create the two vectors AB and AC.

AB = = <-1-1, 1-2, 1+1> = <-2, -1, 2>
AC = = <-3-1, 2-2, 1+1> = <-4, 0, 2>

The area of the triangle is equal to 1/2 the magnitude of the cross product.

AB X AC = <-2, -1, 2> X <-4, 0, 2> = <-2, -4, -4>

|| AB X AC || = √[(-2)² + (-4)² + (-4)²] = √(4 + 16 + 16)
|| AB X AC || = √36 = 6

The area of the triangle is

Area = (1/2)*6 = 3