 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/c/solution

D = a \cdot (b \times c) > 0

. Since these vectors form a basis, write

r = α a + β b + γ c

α = \frac{r \cdot ( b \times c )}{D}, β = \frac{r \cdot ( c \times a )}{D}, γ = \frac{r \cdot ( a \times b )}{D} .

A point is in the closed tetrahedron precisely when it is a convex combination of the four vertices, with weights

1 - α - β - γ, α, β, γ

. Thus the necessary and sufficient conditions are

r \cdot (b \times c) \geq 0, r \cdot (c \times a) \geq 0, r \cdot (a \times b) \geq 0, r \cdot (a \times b + b \times c + c \times a) \leq D .

These four half-spaces are the four inward sides of its faces. This is tetrahedron membership via scalar triple products; making all inequalities strict describes the interior.

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