Scalar Triple Vector Product
When solving physics problems, we often calculate the vector multiplication with three vectors. Herein, we consider three vectors: \begin{eqnarray} {\bf A} \cdot ({\bf B} \times {\bf C} )\ \ \ \ \ \ \ \ \ \ \ \ (1) \end{eqnarray} The calculation formula as shown above can be solved as follows: \begin{eqnarray} {\bf A} \cdot ({\bf B} \times {\bf C} ) = {\bf C} \cdot ({\bf A} \times {\bf B} ) = {\bf B} \cdot ({\bf C} \times {\bf A} )\ \ \ \ \ \ \ \ \ \ \ \ (2) \end{eqnarray} This is called the scalar triple vector product. Herein, I intend to explain the scalar triple vector product.The cross product between the vector \( {\bf A} \) and the vector \( {\bf B} \) is a vector, having \(x,\ y\), and \(z\) components. The cross product between the vector \( {\bf A} \) and the vector \( {\bf B} \) can be expressed by the determinant, as follows: \begin{eqnarray} {\bf A} \times {\bf B} = \left| \begin{array}{ccc} {\bf i} & {\bf j} & {\bf k} \\ A_x & A_y & A_z \\ B_x & B_y & B_z \end{array} \right| \end{eqnarray} Thus, the \(x,\ y\), and \(z\) components of the \({\bf A} \times {\bf B}\) are written as follows: \begin{eqnarray} (A_yB_z - A_zB_y,\ A_zB_x - A_xB_z,\ A_xB_y - A_yB_x) \end{eqnarray} Thus, the components of \({\bf A}\cdot({\bf B} \times {\bf C})\) are: \begin{eqnarray} {\bf A} \cdot ({\bf B} \times {\bf C} ) &=& A_x (B_yC_z - B_zC_y) + A_y(B_zC_x - B_xC_z) + A_z (B_xC_y - B_yC_x) \\ &=& A_x B_y C_z - A_xB_zC_y + A_yB_zC_x - A_yB_xC_z + A_zB_xC_y - A_zB_yC_x \end{eqnarray} The vectors A, B, and C are cyclically replaced. Thus, we obtain: \begin{eqnarray} {\bf C} \cdot ({\bf A} \times {\bf B} ) &=& C_x (A_yB_z - A_zB_y) + C_y(A_zB_x - A_xB_z) + C_z (A_xB_y - A_yB_x) \\ &=& C_x A_y B_z - C_xA_zB_y + C_yA_zB_x - C_yA_xB_z + C_zA_xB_y - C_zA_yB_x &=& {\bf A} \cdot ({\bf B} \times {\bf C} ) \end{eqnarray} \( {\bf B} \cdot ({\bf C} \times {\bf A})\) has a similar result, as follows: \begin{eqnarray} {\bf A} \cdot ({\bf B} \times {\bf C} ) = {\bf C} \cdot ({\bf A} \times {\bf B} ) = {\bf B} \cdot ({\bf C} \times {\bf A} ) \end{eqnarray} Thus, the scalar triple product will be equal, even if the vectors are cyclically replaced.
The cross product of three vectors, as presented below, is called the vector triple product. \begin{eqnarray} {\bf A} \times ( {\bf B} \times {\bf C} ) = ({\bf C}\cdot {\bf A}){\bf B} - ({\bf A}\cdot {\bf B}){\bf B} \end{eqnarray} This will be described in “Vector triple product” page.
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