 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2016/paper-2/11/b/solution

Take two edge vectors in the face,

b - a

and

c - a

n = (b - a) \times (c - a) = a \times b + b \times c + c \times a .

The face plane is

(r - a) \cdot n = 0

. Every term in

a \cdot n

except

a \cdot (b \times c)

vanishes, giving

r \cdot (a \times b + b \times c + c \times a) = a \cdot (b \times c) .

The scalar triple product is nonzero, so this normal cannot vanish.

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