Cross product commutativity

Question

I came across the definition of cross products, which:

$a \times b = |a||b| \sin \theta$

But I have also noticed that $b \times a = -(a \times b)$

I do not really follow, isn't

$b \times a = |b||a|\sin \theta$?

Clearly not.

Answer 1

You're missing one part in your cross product formula: the cross product is actually $$\mathbf a\times \mathbf b = |\mathbf a||\mathbf b|\sin(\theta)\hat {\mathbf n}$$

And that's where your confusion is coming from because it's exactly that $\hat {\mathbf n}$ that is changing signs whenever you take the cross product in the alternate order.

$\hat {\mathbf n}$ is the right-handed unit normal to the plane containing the vectors $\mathbf a$ and $\mathbf b$. The term "right-handed" here means that if you were to place the side of your right hand (the part you'd karate chop someone with) parallel to the first vector in the cross product, $\mathbf a$, and curl your finger toward the second vector in the cross product, $\mathbf b$, then your thumb would point in the direction of the right-handed normal.

Using your right hand, confirm for yourself that the directions of $\mathbf a\times \mathbf b$ and $\mathbf b \times \mathbf a$ are opposite. Thus the negative sign in the identity $$\mathbf a \times \mathbf b = -\mathbf b \times \mathbf a$$