Are the four points A(1,−1,1), B(−1,1,1), C(1,1,1) and D(2,−3,4) coplanar? Justify your answer.

To determine if the points A(1,−1,1), B(−1,1,1), C(1,1,1), and D(2,−3,4) are coplanar, we can use the concept of vectors and the scalar triple product.

Step 1: Create Vectors

First, we need to create vectors from these points. We can form three vectors using one point as a reference. Let's use point A:

• Vector AB = B - A = (−1 - 1, 1 - (−1), 1 - 1) = (−2, 2, 0)
• Vector AC = C - A = (1 - 1, 1 - (−1), 1 - 1) = (0, 2, 0)
• Vector AD = D - A = (2 - 1, −3 - (−1), 4 - 1) = (1, −2, 3)

Step 2: Calculate the Scalar Triple Product

The points are coplanar if the scalar triple product of vectors AB, AC, and AD is zero. The scalar triple product can be calculated using the determinant of a matrix formed by these vectors:

Let:

| -2  2  0 |
|  0  2  0 |
|  1 -2  3 |

Step 3: Compute the Determinant

The determinant can be calculated as follows:

= -2(2*3 - 0*(-2)) - 2(0*3 - 0*1) + 0(0*(-2) - 2*1)
= -2(6) + 0 + 0
= -12

Final Assessment

Since the scalar triple product is not zero (it is -12), the points A, B, C, and D are not coplanar. Thus, they do not lie on the same plane.