Scalar Triple Product⚠ unverified

Understanding the Parameters

• a_x (m) — Length.
• a_y (m) — Length.
• a_z (m) — Length.
• b_x (m) — Length.
• b_y (m) — Length.
• b_z (m) — Length.
• c_x (m) — Length.
• c_y (m) — Length.
• c_z (m) — Length.
• Output [a,b,c] (m^3) — Signed volume.

How to Calculate

1. Enter Length as ax (default 1 m). Use the unit menu when you need a different unit.
2. Enter Length as ay (default 0 m). Use the unit menu when you need a different unit.
3. Enter Length as az (default 0 m). Use the unit menu when you need a different unit.
4. Enter Length as bx (default 0 m). Use the unit menu when you need a different unit.
5. Enter Length as by (default 1 m). Use the unit menu when you need a different unit.
6. Enter Length as bz (default 0 m). Use the unit menu when you need a different unit.
7. Enter Length as cx (default 0 m). Use the unit menu when you need a different unit.
8. Enter Length as cy (default 0 m). Use the unit menu when you need a different unit.
9. Enter Length as cz (default 1 m). Use the unit menu when you need a different unit.
10. Click Calculate. The card evaluates $[a,b,c]=a\cdot(b\times c)$ and shows the result in the declared output unit; Result in (when present) converts that number.

The Science: A Rigorous Derivation (Approaching a Proof)

A Euclidean vector is an arrow: magnitude and direction. The dot product $\mathbf{a}\cdot\mathbf{b} = |a||b|\cos\theta$ is the unique symmetric bilinear form inducing the Euclidean norm. The cross product in $\mathbb{R}^3$ is the unique alternating bilinear map (up to scale) whose magnitude is the parallelogram area and whose direction follows the right-hand rule. The scalar triple product $\mathbf{a}\cdot(\mathbf{b}\times\mathbf{c})$ is the signed volume of the parallelepiped, and vanishes iff the three vectors are coplanar.

Dimensional check. The declared output unit is m^3; ToolBox evaluates the formula in SI (radians internally for every trigonometric call) and python-calc converts to the unit on the card.

History and Development

• Gibbs (1881 notes, 1901 textbook) and Heaviside. The modern $\mathbf{a}\cdot\mathbf{b}$ and $\mathbf{a}\times\mathbf{b}$ notation, distilled from Hamilton's quaternions.

• Grassmann, Ausdehnungslehre (1844). Exterior algebra — the cross product is a $2$-vector in $3$ dimensions identified with a vector via the Hodge star.

• Hamilton, quaternions (1843). $ij=k$ is the cross product plus a scalar (the negative of the dot product). Gibbs/Heaviside split that into two operations engineers actually use.

Related Concepts: Vector Magnitude 2D, Vector Magnitude 3D, Dot Product, Angle Between Vectors, Cross Product Magnitude, Scalar Projection, Direction Cosine