Scalar Triple Product

The scalar triple product is an operation performed on three vectors in three-dimensional space, which results in a scalar quantity. It is a measure of the volume of the parallelepiped formed by the three vectors. The scalar triple product is also known as the box product or mixed product.

Definition

Given three vectors $\vec{a}$, $\vec{b}$, and $\vec{c}$ in $\mathbb{R}^3$, the scalar triple product is defined as:

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

Here, $\vec{b} \times \vec{c}$ is the cross product of vectors $\vec{b}$ and $\vec{c}$, which results in a vector that is perpendicular to the plane containing $\vec{b}$ and $\vec{c}$. The dot product of $\vec{a}$ with this new vector gives the scalar triple product.

Geometric Interpretation

The absolute value of the scalar triple product $|[\vec{a} \vec{b} \vec{c}]|$ gives the volume of the parallelepiped formed by the vectors $\vec{a}$, $\vec{b}$, and $\vec{c}$ as adjacent edges. If the scalar triple product is zero, it means that the three vectors are coplanar, and the volume of the parallelepiped is zero.

Properties

The scalar triple product has several important properties:

Property	Description
Antisymmetry	$[\vec{a} \vec{b} \vec{c}] = -[\vec{a} \vec{c} \vec{b}] = -[\vec{b} \vec{a} \vec{c}]$
Cyclic Permutation	$[\vec{a} \vec{b} \vec{c}] = [\vec{b} \vec{c} \vec{a}] = [\vec{c} \vec{a} \vec{b}]$
Linearity	$[\vec{a} (\vec{b} + \vec{d}) \vec{c}] = [\vec{a} \vec{b} \vec{c}] + [\vec{a} \vec{d} \vec{c}]$
Scalar Multiplication	$[k\vec{a} \vec{b} \vec{c}] = k[\vec{a} \vec{b} \vec{c}]$ for any scalar $k$

Calculation

To calculate the scalar triple product, you can use the determinant of a matrix composed of the components of the vectors:

$$ [\vec{a} \vec{b} \vec{c}] = \begin{vmatrix} a_1 & a_2 & a_3 \ b_1 & b_2 & b_3 \ c_1 & c_2 & c_3 \ \end{vmatrix} $$

Here, $a_1, a_2, a_3$ are the components of vector $\vec{a}$, and similarly for $\vec{b}$ and $\vec{c}$.

Examples

Example 1: Calculation of Scalar Triple Product

Let $\vec{a} = \begin{pmatrix} 1 \ 2 \ 3 \end{pmatrix}$, $\vec{b} = \begin{pmatrix} 4 \ 5 \ 6 \end{pmatrix}$, and $\vec{c} = \begin{pmatrix} 7 \ 8 \ 9 \end{pmatrix}$. Calculate the scalar triple product $[\vec{a} \vec{b} \vec{c}]$.

$$ [\vec{a} \vec{b} \vec{c}] = \begin{vmatrix} 1 & 2 & 3 \ 4 & 5 & 6 \ 7 & 8 & 9 \ \end{vmatrix} = 1(5 \cdot 9 - 6 \cdot 8) - 2(4 \cdot 9 - 6 \cdot 7) + 3(4 \cdot 8 - 5 \cdot 7) = 0 $$

Since the scalar triple product is zero, the vectors $\vec{a}$, $\vec{b}$, and $\vec{c}$ are coplanar.

Example 2: Geometric Interpretation

Consider vectors $\vec{a} = \begin{pmatrix} 1 \ 0 \ 0 \end{pmatrix}$, $\vec{b} = \begin{pmatrix} 0 \ 1 \ 0 \end{pmatrix}$, and $\vec{c} = \begin{pmatrix} 0 \ 0 \ 1 \end{pmatrix}$. The scalar triple product $[\vec{a} \vec{b} \vec{c}]$ represents the volume of the parallelepiped formed by these vectors.

$$ [\vec{a} \vec{b} \vec{c}] = \begin{vmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \ \end{vmatrix} = 1 $$

The volume of the parallelepiped is 1, which corresponds to the volume of a unit cube, as expected from the given vectors.

Conclusion

The scalar triple product is a valuable tool in vector calculus and geometry, providing a way to determine the volume of a parallelepiped and to check if vectors are coplanar. Understanding its properties and knowing how to calculate it using determinants are essential skills for students and professionals working with vectors in three-dimensional space.