 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2023/paper-2/11/c/ii/solution

The three displacements from

Q_{1}

are

u = (1, 1, 1)^{T}, v = (- 1, 2, 1)^{T}, w = (0, 3, 2)^{T} .

Here

u \times v = (- 1, - 2, 3)^{T}

and its dot product with

w

is zero. Because the cross product itself is nonzero, the first three points determine a plane and the fourth lies on it. Explicitly all four satisfy

- x - 2 y + 3 z = 3 .

Thus the four points are coplanar.

 New to topics? Read the docs here!