Triple Scalar Product

Definition

The triple scalar product $(\mathbf{u} \times \mathbf{v}) \cdot \mathbf{w}$ is a combination of the cross and dot products that yields a scalar. It represents the signed volume of the parallelepiped defined by the three vectors $\mathbf{u}, \mathbf{v},$ and $\mathbf{w}$.

• How to read: “The cross product of vector u and vector v, dotted with vector w.”
• Meaning: A single number encoding how much 3D volume the three vectors span; sign indicates orientation (right- vs left-handed ordering).

Why It Matters

The triple scalar product provides a fast way to calculate the volume of a parallelepiped (a skewed 3D box) spanned by three vectors. It is a critical tool in crystallography and structural mechanics to determine the volumetric capacity of vector-defined spaces.

Core Concepts

• Determinant Calculation: It can be computed as the determinant of a $3 \times 3$ matrix formed by the components of the three vectors: $(\mathbf{u} \times \mathbf{v}) \cdot \mathbf{w} = \begin{vmatrix} u_1 & u_2 & u_3 \\ v_1 & v_2 & v_3 \\ w_1 & w_2 & w_3 \end{vmatrix}$
  • How to read: “The cross product of vector u and vector v, dotted with vector w, equals the determinant of the matrix formed by the components of u, v, and w.”
  • Meaning: Row-expand this determinant when you have component forms; equivalent to the cross-then-dot definition.
• Volume of Parallelepiped: The volume $V$ is given by the absolute value of the triple scalar product: $V = |(\mathbf{u} \times \mathbf{v}) \cdot \mathbf{w}|$.
  • How to read: “V equals the absolute value of (u cross v) dot w.”
  • Meaning: Physical volume is always non-negative; take absolute value to discard orientation sign.
• Coplanarity Check: If the triple scalar product is zero, the three vectors are coplanar, meaning they lie in the same plane and span zero volume.

Connected Concepts

• Product-to-Sum Formulas
• Sum-to-Product Formulas
• Cross Product
• The Product Rule