Scalar Triple Product

Definition 6.4. For three vectors $\vec a,\vec b,\vec c$, the scalar $(\vec a\times\vec b)\cdot\vec c$ is called the scalar triple product of $\vec a,\vec b,\vec c$.

Geometric meaning. The absolute value $\left|(\vec a\times\vec b)\cdot\vec c\right|$ is the volume of the parallelepiped with $\vec a,\vec b,\vec c$ as coterminous edges. Reason: $|\vec a\times\vec b|$ is the area of the base parallelogram (spanned by $\vec a,\vec b$), and if $\theta$ is the angle between $\vec a\times\vec b$ and $\vec c$, then $|\vec c\,||\cos\theta|$ is the perpendicular height of the parallelepiped above that base; multiplying area $\times$ height gives the volume.

Theorem 6.1 (determinant formula). For $\vec a=a_1\hat i+a_2\hat j+a_3\hat k$, $\vec b=b_1\hat i+b_2\hat j+b_3\hat k$, $\vec c=c_1\hat i+c_2\hat j+c_3\hat k$: …