Question

If vector A=Axi+Ayj+Azk,
B=Bxi+Byj+Bzk and
C=Cxi+Cyj+Czk
Show that the scalar triple product of the vector is[Ax Ay Az][Bx By Bz ][Cx Cy Cz]
Note:the bracket above is a 3 by 3 matrix

Answer 1

Answer on Question #85462, Physics / Other

If vector $\mathbf{A} = A_{x}\mathbf{i} + A_{y}\mathbf{j} + A_{z}\mathbf{k}$, $\mathbf{B} = B_{x}\mathbf{i} + B_{y}\mathbf{j} + B_{z}\mathbf{k}$ and $\mathbf{C} = C_{x}\mathbf{i} + C_{y}\mathbf{j} + C_{z}\mathbf{k}$.

Show that the scalar triple product of the vector is $[A_x A_y A_z][B_x B_y B_z ][C_x C_y C_z]$.

Solution:

The vector product by definition

$$[ \mathbf {B} \times \mathbf {C} ] = \left| \begin{array}{c c c} \mathbf {i} & \mathbf {j} & \mathbf {k} \\ B _ {x} & B _ {y} & B _ {z} \\ C _ {x} & C _ {y} & C _ {z} \end{array} \right| = \mathbf {i} \big (B _ {y} C _ {z} - B _ {z} C _ {y} \big) + \mathbf {j} (B _ {z} C _ {x} - B _ {x} C _ {z}) + \mathbf {k} (B _ {x} C _ {y} - B _ {y} C _ {x})$$

The scalar product by definition

$$(\mathbf {A} \cdot \mathbf {B}) = A _ {x} B _ {x} + A _ {y} B _ {y} + A _ {z} B _ {z}$$

So, for the triple product we obtain

$$\begin{array}{l} (\mathbf {A} \cdot [ \mathbf {B} \times \mathbf {C} ]) = A _ {x} \big (B _ {y} C _ {z} - B _ {z} C _ {y} \big) + A _ {y} (B _ {z} C _ {x} - B _ {x} C _ {z}) + A _ {z} \big (B _ {x} C _ {y} - B _ {y} C _ {x} \big) \\ = \left| \begin{array}{c c c} A _ {x} & A _ {y} & A _ {z} \\ B _ {x} & B _ {y} & B _ {z} \\ C _ {x} & C _ {y} & C _ {z} \end{array} \right| \\ \end{array}$$

Answer provided by https://www.AssignmentExpert.com