Question

what is the cross product A >< B under mirror reflection

Answer 1

Answer on Question #39603, Physics, Mechanics | Kinematics | Dynamics

Question:

what is the cross product $A >< B$ under mirror reflection

Answer:

If a point of an object has coordinates $(x, y, z)$ then the image of this point (as reflected by a mirror in the $y$, $z$ plane) has coordinates $(-x, y, z)$.

Suppose $\vec{A} \times \vec{B} = \vec{C}$

Therefore, cross product transforms:

$$\overrightarrow{A'} \times \overrightarrow{B'} = \left| \begin{array}{ccc} \vec{i} & \vec{j} & \vec{k} \\ -A_x & A_y & A_z \\ -B_x & B_y & B_z \end{array} \right| = \vec{i} C_x - \vec{j} C_y - \vec{k} C_z = \begin{pmatrix} C_x \\ -C_y \\ -C_z \end{pmatrix}$$

Answer: $\overrightarrow{A'} \times \overrightarrow{B'} = \begin{pmatrix} (\vec{A} \times \vec{B})_x \\ -(\vec{A} \times \vec{B})_y \\ -(\vec{A} \times \vec{B})_z \end{pmatrix}$