Scalar Triple Product

The scalar triple product is an operation involving three vectors in three-dimensional space. It results in a scalar quantity that is a measure of the volume of the parallelepiped formed by the three vectors. The scalar triple product is also known as the "mixed product" because it combines the dot product and the cross 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 $\vec{c}$. Then, $\vec{a} \cdot (\vec{b} \times \vec{c})$ is the dot product of vector $\vec{a}$ with the result of the cross product.

Properties

The scalar triple product has several important properties:

Property	Description
Geometric Interpretation	The absolute value of the scalar triple product gives the volume of the parallelepiped formed by the vectors $\vec{a}$, $\vec{b}$, and $\vec{c}$.
Significance of Sign	The sign of the scalar triple product indicates the orientation of the vectors. A positive value means that $\vec{a}$, $\vec{b}$, and $\vec{c}$ form a right-handed system, while a negative value indicates a left-handed system.
Sensitivity to Order	The scalar triple product changes sign when the order of any two vectors is swapped: $[\vec{a} \vec{b} \vec{c}] = -[\vec{a} \vec{c} \vec{b}]$.
Invariance under Cyclic Permutations	The value of the scalar triple product remains the same under cyclic permutations of the vectors: $[\vec{a} \vec{b} \vec{c}] = [\vec{b} \vec{c} \vec{a}] = [\vec{c} \vec{a} \vec{b}]$.

Formula

The scalar triple product can also be expressed in terms of the components of the vectors:

$$ [\vec{a} \vec{b} \vec{c}] = \begin{vmatrix} a_x & a_y & a_z \ b_x & b_y & b_z \ c_x & c_y & c_z \ \end{vmatrix} $$

This is the determinant of a $3 \times 3$ matrix formed by the components of the vectors $\vec{a}$, $\vec{b}$, and $\vec{c}$.

Examples

Example 1: Calculation of Scalar Triple Product

Given three vectors $\vec{a} = \langle 1, 2, 3 \rangle$, $\vec{b} = \langle 4, 5, 6 \rangle$, and $\vec{c} = \langle 7, 8, 9 \rangle$, find the scalar triple product $[\vec{a} \vec{b} \vec{c}]$.

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

The result is zero, which indicates that the vectors are coplanar, and the volume of the parallelepiped is zero.

Example 2: Geometric Interpretation

Consider vectors $\vec{a} = \langle 1, 0, 0 \rangle$, $\vec{b} = \langle 0, 1, 0 \rangle$, and $\vec{c} = \langle 0, 0, 1 \rangle$. The scalar triple product is:

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

The result is one, which is the volume of the unit cube formed by the standard basis vectors in $\mathbb{R}^3$.

Conclusion

The scalar triple product is a valuable tool in vector calculus and physics, providing insights into the spatial relationships between vectors. It is essential for understanding concepts such as volume, orientation, and coplanarity in three-dimensional space. By mastering the properties and calculations of the scalar triple product, students can solve complex problems in geometry and vector analysis.