The triple scalar product

Calculating it

a · (b × c) is the 3 × 3 determinant with rows a, b and c:

| a1 a2 a3 |

| b1 b2 b3 |

| c1 c2 c3 |

Expand along the top row or find b × c first and dot with a.

Volumes

• Parallelepiped with edges a, b, c from one corner: volume = |a · (b × c)|.
• Tetrahedron with edges a, b, c from one vertex: volume = (1/6)|a · (b × c)|.

For a tetrahedron ABCD use the edges AB, AC and AD from vertex A.

Properties

• Cyclic: a · (b × c) = b · (c × a) = c · (a × b). Swapping two vectors changes the sign.
• a · (b × c) = 0 when a, b and c are coplanar (the box is flat), or when any two are parallel.
• The sign tells you whether a, b, c form a right-handed set (positive) or a left-handed set (negative).

Worked example

Find the volume of the tetrahedron with vertices A(1, 1, 0), B(3, 2, 1), C(2, 4, 2) and D(1, 2, 5).

1. AB = (2, 1, 1), AC = (1, 3, 2), AD = (0, 1, 5)
2. Determinant = 2(15 - 2) - 1(5 - 0) + 1(1 - 0) = 26 - 5 + 1 = 22
3. Volume = 22/6

Answer: 11/3

Check yourself

1. Find the volume of the parallelepiped with edges a = i + 2k, b = 2i + j and c = 3j + k.
   Show answer

   13. Determinant: 1(1×1 - 0×3) - 0 + 2(2×3 - 1×0) = 1 + 12 = 13.

2. Find the volume of the tetrahedron with vertices A(1, 1, 0), B(3, 2, 1), C(2, 4, 2) and D(1, 2, 5). Give your answer as a fraction in its simplest form.
   Show answer

   11/3. AB = (2, 1, 1), AC = (1, 3, 2), AD = (0, 1, 5). The determinant is 22, so the volume is 22/6 = 11/3.

3. The vectors i + 2j + 3k, 2i + pj + k and i - j + 2k are coplanar, where p is a constant. Find p.
   Show answer

   -11. Determinant: 1(2p + 1) - 2(4 - 1) + 3(-2 - p) = 2p + 1 - 6 - 6 - 3p = -p - 11 = 0, so p = -11.

4. Which is equal to a · (b × c)?
   • b · (a × c)
   • b · (c × a)
   • a · (c × b)
   • (a × b) × c
   Show answer

   b · (c × a). Cycling the vectors (a to b to c to a) keeps the value; swapping two of them changes the sign.

5. a · (b × c) = 0, with a, b and c non-zero. Which statement must be true?
   • a, b and c are mutually perpendicular
   • a is parallel to b × c
   • a, b and c have equal lengths
   • a, b and c lie in one plane
   Show answer

   a, b and c lie in one plane. The parallelepiped has zero volume, so it is flat: the three vectors are coplanar.

6. O is the origin, A(2, 0, 0), B(0, 3, 0) and C(0, 0, 4). Find the volume of the tetrahedron OABC.
   Show answer

   4. a · (b × c) = 2 × 3 × 4 = 24, so the volume is 24/6 = 4.

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Sources

• Pearson Edexcel International Advanced Level in Mathematics, Further Mathematics and Pure Mathematics specification (issue 3), Unit FP3

Last updated 8 November 2026. Spotted a mistake? Tell us.