why does the second term of the cross product ( in the direction of $y$) have minus before it? Thanks in advance
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$\begingroup$It is necessary to fulfill the definition of the cross-product.
From the geometrical point of view, since cross-product corresponds to the signed area of the parallelogram which has the two vectors as sides, we can find the minus-sign in its expression by the symbolic determinant which indeed requires a minus-sign for the $\vec j$ coordinate, according to Laplace’s expansion for the determinant.
This obviously is not proof but it’s a good way to remember that.
To prove, we can start by the definition for the unitary vectors along coordinates axes:
$\vec i \times \vec j=\vec k, \quad \vec j \times \vec k=\vec i, \quad \vec k\times \vec i=\vec j.$
$\vec j \times \vec i=-\vec k, \quad \vec k \times \vec j=-\vec i, \quad \vec i\times \vec k=-\vec j$.
And assuming
- $\vec u = u_x \vec i +u_y \vec j +u_z \vec k$;
- $\vec v = v_x \vec i +v_y \vec j +v_z \vec k$.
And then computing the product using the given definitions for the unitary vectors.
$\endgroup$ $\begingroup$That is how the cross product is defined.
It is defined this way to ensure, for example, that $\mathbf{a}\times\mathbf{b}$ is perpendicular to $\mathbf{a}$ and $\mathbf{b}$.
$\endgroup$ $\begingroup$The cross product has an associated sign depending on the order of constitutive vectors it is formed out of. Accordingly the sign must be compulsorily required to be defined with a rule, the right hand rule.
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