Scalar Triple Product Calculator
Compute a·(b×c) for three 3D vectors and the volume of the parallelepiped they span.
Equals the signed determinant det[a; b; c].
- 1
Cross product b×c: i
b₂·c₃ − b₃·c₂ = 2·3 − 0·0 = 6 - 2
Cross product b×c: j
b₃·c₁ − b₁·c₃ = 0·0 − 0·3 = 0 - 3
Cross product b×c: k
b₁·c₂ − b₂·c₁ = 0·0 − 2·0 = 0 - 4
Scalar triple product a·(b×c)
1·6 + 0·0 + 0·0 = 6Absolute value gives parallelepiped volume; sign shows orientation.
How does this calculator work?
The scalar triple product a·(b×c) multiplies vector a by the cross product of b and c, giving a single number equal to the determinant det[a; b; c]. Its absolute value is the volume of the parallelepiped spanned by the three vectors, and its sign reveals their handedness; zero means they are coplanar.
Formula
How this is calculated
Enter three 3D vectors a, b, and c, each as three components separated by commas or spaces. The calculator first forms the cross product b×c = (b₂c₃−b₃c₂, b₃c₁−b₁c₃, b₁c₂−b₂c₁), a vector perpendicular to both b and c whose magnitude equals the area of the parallelogram they span.
The scalar triple product is then the dot product a·(b×c) = a₁(b×c)₁ + a₂(b×c)₂ + a₃(b×c)₃. This single number is identical to the determinant of the 3×3 matrix whose rows (or columns) are a, b, and c, so it is symmetric under cyclic permutation: a·(b×c) = b·(c×a) = c·(a×b).
Geometrically the absolute value |a·(b×c)| is the volume of the parallelepiped with edges a, b, and c. The sign indicates orientation: positive for a right-handed triple, negative for left-handed. A result of zero means the three vectors are linearly dependent (coplanar) and the parallelepiped is flat with zero volume. All inputs must have exactly three numeric components; the output carries the same units cubed as the inputs.
Frequently asked questions
It means the three vectors are coplanar (linearly dependent), so the parallelepiped they form is degenerate and has zero volume.
The scalar triple product is signed. A positive value indicates a right-handed orientation of a, b, c; a negative value indicates a left-handed orientation. The volume is the absolute value.
Yes. The scalar triple product equals the determinant of the matrix formed by placing the three vectors as its rows (or columns).
Also known as
TG we-Calculate Editorial Team. (2026). Scalar Triple Product Calculator [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/scalar-triple-product-calculator
TG we-Calculate Editorial Team. "Scalar Triple Product Calculator." TG we-Calculate. 2026. https://we-calculate.com/calculator/scalar-triple-product-calculator.
TG we-Calculate Editorial Team, "Scalar Triple Product Calculator," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/scalar-triple-product-calculator
@misc{wecalculate_scalar_triple_product_calculator, title = {Scalar Triple Product Calculator}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/scalar-triple-product-calculator}}, year = {2026}, note = {TG we-Calculate} }
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