AcademyVectors In Rn
Academy
Scalar Triple Product
Level 1 - Linear Algebra topic page in Vectors In Rn.
Principle
The scalar triple product combines three vectors in \(\mathbb R^3\) and returns a scalar. Its absolute value is the volume of the parallelepiped formed by the three vectors.
Algebraically, it is a dot product after a cross product.
Notation
\(\mathbf u\cdot(\mathbf v\times\mathbf w)\)
the scalar triple product of \mathbf u, \mathbf v, and \mathbf w
\(|\mathbf u\cdot(\mathbf v\times\mathbf w)|\)
volume of the parallelepiped spanned by the three vectors
\(\det A\)
the determinant of a square matrix A
\(\mathbf 0\)
the zero vector
The Core Test
Compute the scalar triple product by first taking a cross product, then a dot product:
Scalar triple product
\[[\mathbf u,\mathbf v,\mathbf w]=\mathbf u\cdot(\mathbf v\times\mathbf w)\]
Equivalently, place the three vectors as rows or columns of a \(3\times3\) matrix and take the determinant:
Determinant form
\[\mathbf u\cdot(\mathbf v\times\mathbf w)=\det\begin{pmatrix}u_1&u_2&u_3\\v_1&v_2&v_3\\w_1&w_2&w_3\end{pmatrix}\]
The geometric tests are:
- \(|\mathbf u\cdot(\mathbf v\times\mathbf w)|\) is volume.
- \(\mathbf u\cdot(\mathbf v\times\mathbf w)=0\) means the three vectors are coplanar.
- Reversing two inputs changes the sign but not the volume.
Worked Cases
Question
Compute \(\mathbf u\cdot(\mathbf v\times\mathbf w)\) for \(\mathbf u=(1,2,0)\), \(\mathbf v=(0,1,3)\), and \(\mathbf w=(2,-1,1)\).
Answer
First compute \(\mathbf v\times\mathbf w\). Using the formula, \((0,1,3)\times(2,-1,1)=(1\cdot1-3(-1),3\cdot2-0\cdot1,0(-1)-1\cdot2)=(4,6,-2)\). Now dot with \(\mathbf u\): \((1,2,0)\cdot(4,6,-2)=1\cdot4+2\cdot6+0\cdot(-2)=16\). The scalar triple product is \(16\).
Examples
Question
What does \(\mathbf u\cdot(\mathbf v\times\mathbf w)=0\) tell you?
Answer
The volume of the parallelepiped is \(|\mathbf u\cdot(\mathbf v\times\mathbf w)|\). If the scalar triple product is zero, the volume is zero. That means the three vectors lie in a common plane or one of them is the zero vector, so they do not span three-dimensional volume.