Scalar Triple Product Calculator
Written by Thierno Sadou Diallo, formula verified per our methodology • Last checked on 9/10/2026
The scalar triple product of three vectors a, b, and c is calculated with a · (b × c), and gives the signed volume of the parallelepiped they form. For the three basis vectors (1,0,0), (0,1,0), and (0,0,1), which form a unit cube, the scalar triple product is exactly 1.
Explanation
The scalar triple product combines the two fundamental vector operations already on this site: it is the dot product of the first vector with the cross product of the other two, a · (b × c). Geometrically, its absolute value gives exactly the volume of the parallelepiped (a box with faces parallel in pairs, but not necessarily rectangular) formed by the three vectors placed from a common origin — a result directly verifiable on an orthogonal box: the vectors (2,0,0), (0,3,0), and (0,0,4) give a scalar triple product of 24, exactly the volume of a cuboid of dimensions 2×3×4. The sign of the result carries additional geometric information: a positive scalar triple product indicates that the three vectors, taken in that order, form a right-handed frame; a negative result indicates a left-handed frame. A scalar triple product of exactly zero has a particular and very useful meaning in practice: it indicates that the three vectors are coplanar (they all lie in the same plane), since no three-dimensional volume can then be formed — this is in fact the most direct test to check whether three vectors (or three points, once converted to vectors) are coplanar.
Example: orthogonal box, vectors (2,0,0), (0,3,0), (0,0,4)
Inputs
a = (2, 0, 0). b = (0, 3, 0). c = (0, 0, 4).
Calculation
b × c = (3×4−0×0, 0×0−0×4, 0×0−3×0) = (12, 0, 0). a · (b × c) = 2×12 + 0×0 + 0×0 = 24.
Result
The scalar triple product is 24, exactly the volume of the 2×3×4 box they bound.
Frequently asked questions
What does a zero scalar triple product mean?
A zero scalar triple product means the three vectors are coplanar: they all lie in the same plane and therefore cannot form a three-dimensional volume, as in the test case of the vectors (1,0,0), (0,1,0), and (1,1,0), which all belong to the xy plane. It is the most direct method to algebraically test whether three vectors (or three points in space) are coplanar.
What does a negative scalar triple product mean?
A negative result is not an error: it simply indicates that the three vectors, taken in the order a, b, c, form a left-handed frame rather than a right-handed one. To get only the volume of the parallelepiped, regardless of this orientation, this calculator also displays the absolute value of the scalar triple product.
How does this calculation relate to the dot product and cross product already on the site?
The scalar triple product is nothing but the combination of the two: you first calculate the cross product of b and c (which gives a new vector, perpendicular to the plane formed by b and c, whose magnitude is the area of the parallelogram they bound), then the dot product of that result with the vector a. It is this two-step combination that turns a 2D area into a 3D volume.