The scalar triple product of three vectors $\vec{A}$, $\vec{B}$, and $\vec{C}$ is defined as the dot product of one vector with the cross product of the other two. It results in a scalar quantity.

Geometric Interpretation

Geometrically, the absolute value of the scalar triple product represents the volume of the parallelepiped spanned by the three vectors $\vec{A}$, $\vec{B}$, and $\vec{C}$.

Properties

• Cyclic Permutation: The scalar triple product is invariant under cyclic permutation of the vectors.
  $$\vec{A} \cdot (\vec{B} \times \vec{C}) = \vec{B} \cdot (\vec{C} \times \vec{A}) = \vec{C} \cdot (\vec{A} \times \vec{B})$$
• Determinant Form: It can be easily calculated as the determinant of a $3 \times 3$ matrix formed by the components of the three vectors.
  $$\vec{A} \cdot (\vec{B} \times \vec{C}) = \begin{vmatrix} A_x & A_y & A_z \\ B_x & B_y & B_z \\ C_x & C_y & C_z \end{vmatrix}$$
• Coplanar Vectors: If the scalar triple product is zero, the three vectors are coplanar (they lie in the same plane), meaning the volume of the parallelepiped they form is zero.

The vector triple product is the cross product of one vector with the cross product of two others. It results in a vector.

The resulting vector lies in the plane formed by vectors $\vec{B}$ and $\vec{C}$. Therefore, it can be expressed as a linear combination of $\vec{B}$ and $\vec{C}$.

BAC-CAB Rule (Lagrange's Formula)

A highly useful identity for expanding the vector triple product is known as the BAC-CAB rule:

Key Properties

• Non-Associativity: The cross product is not associative. In general,
  $$\vec{A} \times (\vec{B} \times \vec{C}) \neq (\vec{A} \times \vec{B}) \times \vec{C}$$
• Jacobi Identity: For any three vectors, the sum of their cyclic vector triple products is zero.
  $$\vec{A} \times (\vec{B} \times \vec{C}) + \vec{B} \times (\vec{C} \times \vec{A}) + \vec{C} \times (\vec{A} \times \vec{B}) = \vec{0}$$