Scalar Triple Product Calculator
Enter three 3-D vectors to compute the scalar triple product a·(b×c), equal to the signed volume of the parallelepiped they span. a=(1,0,0), b=(0,1,0), c=(0,0,1) → 1 (unit cube).
Input Data
Results
At a glance:The scalar triple product a·(b×c) (also a×b·c, the dot of one with the cross of the other two) is a scalar equal to the signed volume of the parallelepiped formed by vectors a, b, c. Geometrically: |a·(b×c)| is the volume of the parallelepiped; the sign is positive or negative depending on the right-/left-handedness of the three vectors. Key properties: (1) it is zero exactly when a, b, c are coplanar (linearly dependent); (2) it is alternating — swapping any two vectors flips the sign; (3) the three orderings a·(b×c)=b·(c×a)=c·(a×b) are equal (cyclic). It is computed by the determinant of the 3×3 component matrix — expand along the first row. Example: a=(1,0,0), b=(0,1,0), c=(0,0,1) → b×c=(1,0,0), a·(b×c)=1 (unit-cube volume). Applications: (1) volume of a parallelepiped or tetrahedron (=⅙|triple|); (2) coplanarity test (triple=0 means the three vectors lie in a plane); (3) linear-dependence test in vector geometry; (4) crystallography (unit-cell volume); (5) computing the distance from a point to a plane via the signed volume.
Formula
Scalar triple product: V = a·(b×c) = det[a b c]
Volume of parallelepiped: |V|
Coplanar test: V = 0 ⇒ a,b,c coplanar
Cyclic: a·(b×c) = b·(c×a) = c·(a×b)
Tetrahedron volume: V_tet = |V| / 6
$$\mathbf{a}\cdot(\mathbf{b}\times\mathbf{c}) = \begin{vmatrix} a_x & a_y & a_z \\ b_x & b_y & b_z \\ c_x & c_y & c_z \end{vmatrix}, \quad V_{parallel} = |\mathbf{a}\cdot(\mathbf{b}\times\mathbf{c})|$$How to Use
- Enter the x, y, z components of vectors a, b, c.
- The tool shows a·(b×c) and its absolute value (the parallelepiped volume).
- A zero result means the three vectors are coplanar.
Case Studies
Parallelepiped and tetrahedron volume
a=(1,0,0), b=(0,1,0), c=(0,0,1) → V=1, unit-cube volume.
A tetrahedron with these edges has volume |V|/6 = 1/6.
If c=(1,1,0) the three lie in the xy-plane, V=0 (coplanar).
Content review: Calculatorism Science Team. Results are for reference only; please refer to the relevant authorities for the official figures.