Question

Construct an example to show that the cross product of two vectors in 3 d does not transform like a vector under a reflection and therefore is a pseudo vector

Answer 1

A pseudovector is a quantity that is similar to a vector but undergoes an additional inversion under a coordinate reflection.

For example, $x\times y=z$

Under a reflection in the xy plane, this equation becomes $x\times y=-z$

example:

$x=(1,2,3),y=(0,1,2)$

$x\times y=\begin{vmatrix} i & j&k \\ 1 & 2&3\\ 0&1&2 \end{vmatrix}=i-2j+k$

$y\times x=\begin{vmatrix} i & j&k \\ 0 & 1&2\\ 1&2&3 \end{vmatrix}=-i+2j-k=-(x\times y)$