2 Cross Products?

Question

Usually, if we want to find the cross product of 2 vectors $\vec{b}$ and $\vec{c}$, we want to find the vector which is perpendicular to both of them. Let's say the cross product of $\vec{b}$ and $\vec{c}$ is $\vec{d}$. Isn't $-\vec{d}$ then also perpendicular to $\vec{b}$ and $\vec{c}$? Does that mean that there are 2 cross products, or am I making a mistake?

Answer 1

The cross product of $\vec b$ and $\vec c$ is defined as the vector with the following properties:

1. The length of the product is equal to $|\vec b|\cdot|\vec c|\cdot\sin(\alpha)$, where $\alpha$ is the angle between the two vectors.
2. The product is perpendicular to both $\vec b$ and $\vec c$.
3. The direction of the product is such that it follows the right hand rule.

The last point ensures that the cross product is uniquelly defined by $b$ and $c$. That is, of the two vectors that satisfy points 1 and 2, only one of them satisfies point 3

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Note that there are many interpretations of the right hand rule, from (literally) hand-wavy ones, to (for the purpose of this question) circular ones (i.e., one way to define the right hand rule would be to say that it is defined by the direction of the cross product).

Let's strike a balance then and define the right hand rule as such:

If $\vec a \times \vec b=\vec c$, then, looking onto the plane, spanned by $\vec a$ and $\vec b$ from the positive side (i.e., from the side into which $\vec c$ points into), the angle required to rotate $\vec a$ into $\vec b$ is smaller than the angle required to rotate $\vec b$ into $\vec a$.