Scalar Triple Product Calculator
The scalar triple product of three 3D vectors a, b, c is the scalar a . (b x c), computed as the dot product of a with the cross product of b and c. This is equivalent to computing the determinant of the 3x3 matrix with a, b, c as its rows. The result has a clear geometric interpretation: its absolute value equals the volume of the parallelepiped (the 3D solid analogous to a parallelogram) with edges defined by the three vectors. If the scalar triple product is zero, the vectors are coplanar. Enter the three vectors below to compute the scalar triple product and parallelepiped volume instantly.
Vector a
Vector b
Vector c
Scalar triple product formula
a.(b x c) = a1(b2*c3 - b3*c2) - a2(b1*c3 - b3*c1) + a3(b1*c2 - b2*c1)
= det([[a1,a2,a3],[b1,b2,b3],[c1,c2,c3]])
Volume = |a.(b x c)|
The formula expands the cross product b x c and then dots with a, which is exactly the cofactor expansion of the 3x3 determinant along the first row. The sign indicates orientation (right-handed vs. left-handed).
Properties of the scalar triple product
- It is zero if and only if the three vectors are linearly dependent (coplanar).
- It is cyclic: a.(b x c) = b.(c x a) = c.(a x b).
- Swapping any two vectors negates the result: a.(b x c) = -a.(c x b).
- It equals the 3x3 determinant with the vectors as rows.
- The parallelepiped volume is always non-negative: Volume = |a.(b x c)|.
Scalar triple product: frequently asked questions
What is the scalar triple product?
The scalar triple product of three 3D vectors a, b, c is defined as a . (b x c): the dot product of a with the cross product of b and c. It is a scalar (a single number) and equals the determinant of the 3x3 matrix whose rows are a, b, and c.
What is the geometric meaning of the scalar triple product?
The absolute value of the scalar triple product gives the volume of the parallelepiped (a 3D figure with six parallelogram faces) formed by the three vectors as edges from a common vertex. If the result is zero, the three vectors are coplanar.
How is the scalar triple product calculated?
First compute the cross product b x c = (b2*c3-b3*c2, b3*c1-b1*c3, b1*c2-b2*c1). Then take the dot product of a with this result: a1*(b2*c3-b3*c2) - a2*(b1*c3-b3*c1) + a3*(b1*c2-b2*c1). This is also the 3x3 determinant expanding along the first row.
Is the scalar triple product commutative?
The scalar triple product is cyclic: a.(b x c) = b.(c x a) = c.(a x b). Swapping any two vectors changes the sign: a.(b x c) = -a.(c x b). So the absolute value (volume) is unchanged by any permutation, but the sign tracks orientation.
What does it mean if the scalar triple product is zero?
A zero scalar triple product means the three vectors are linearly dependent (coplanar). They lie in the same plane, and the parallelepiped they form has zero volume. This is equivalent to the 3x3 determinant formed by the three row vectors being zero.
Official sources
- NIST Digital Library of Mathematical Functions: dlmf.nist.gov.
- NIST, Mathematical topics: nist.gov/topics/mathematics.
Reviewed by the CalculatorHub team, edited by James Graham, 15 June 2026. See our methodology.