Scalar Triple Product

Consider three vectors in 3-D space: $\vec{a}=<a_1,a_2,a_3>$, $\vec{b}=<b_1,b_2,b_3>$, and $\vec{c}=<c_1,c_2,c_3>$.

The scalar triple product of $\vec{a}$, $\vec{b}$, and $\vec{c}$ is $\vec{a}\cdot(\vec{b}$ x $\vec{c})$. Take note of the fact that this is a scalar, not a vector.

Since the dot product is commutative and the cross product is anticommutative, the scalar triple product does not change in magnitude when one permutes the three vectors. We have the following:

$$\vec{a}\cdot(\vec{b} \times \vec{c})=\vec{b}\cdot(\vec{c} \times \vec{a})=\vec{c}\cdot(\vec{a} \times \vec{b})$$ $$\vec{c}\cdot(\vec{b} \times \vec{a})=\vec{b}\cdot(\vec{a} \times \vec{c})=\vec{a}\cdot(\vec{c} \times \vec{b})$$

The two expressions above are opposite in sign as well. Notice that when $\vec{a}$, $\vec{b}$, and $\vec{c}$ are cycled, the product stays the same, while when the vector order is flipped, the product changes sign.

Using components, $$\vec{a}\cdot(\vec{b} \times \vec{c})=a_1b_2c_3-a_1b_3c_2+a_2b_3c_1-a_2b_1c_3+a_3b_1c_2-a_3b_2c_1.$$ This follows directly from the algebraic definitions of the dot product and the cross product. We can use this formula to verify the above equalities regarding the order of $\vec{a}$, $\vec{b}$, and $\vec{c}$.

Perhaps more interesting is the neat geometric interpretation of the scalar triple product.

We know that $\vec{b}$ x $\vec{c}$ produces a vector whose magnitude is equal to the area of the parallelogram determined by $\vec{b}$ and $\vec{c}$. Since $\vec{b}$ and $\vec{c}$ determine a plane, let $\theta$ denote the angle between $\vec{a}$ and this plane.

Since $\vec{b}$ x $\vec{c}$ is perpendicular to the previously mentioned plane, the angle between $\vec{a}$ and $\vec{b}$ x $\vec{c}$ is $90^{\circ}-\theta$.

Computing the scalar triple product, $$\vec{a}\cdot(\vec{b} \times \vec{c})=\left|\vec{a}\right|\left|\vec{b} \times \vec{c}\right|\cos(90^{\circ}-\theta)=\left|\vec{a}\right|\left|\vec{b} \times \vec{c}\right|\sin\theta.$$

Notice that $\left|\vec{b} \times \vec{c}\right|$ is the area of the parallelogram determined by $\vec{b}$ and $\vec{c}$.

Also, the absolute value of $\left|\vec{a}\right|\sin\theta$ is the height that $\vec{a}$ reaches above the plane.

Using simple volume calculations, we see that the absolute value of the scalar triple product $\vec{a}\cdot(\vec{b} \times \vec{c})$ is the volume of the parallelipiped determined by $\vec{a}$, $\vec{b}$, and $\vec{c}$.