Why do we have minus in y-term of cross product?

Question

why does the second term of the cross product ( in the direction of $y$) have minus before it? Thanks in advance

Answer 1

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.

Answer 2

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 $\mathbf{b}$.

Answer 3

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.