Cross product, ortonormal basis

Question

Could you explain to me why for $\{i, \ j, \ k\}$ an orthonormal basis of $\mathbb{R}^3$ we have $i \times j =k, \ \ j \times k = i, \ \ k \times i =j$?

Thank you.

Answer 1

It's how the cross product is defined. Operations are generally things we define to have some nice properties. Even the notion of addition becomes non-intuitive and somewhat unnatural when you start involving irrational numbers.

As far as anti-symmetry, this is why we use the "right hand rule." Align your right hand along the first vector, curl your fingers towards the second. The resulting vector points in the direction of your thumb.

If you pick the second vector first, observe what happens to the direction of your thumb.

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It's also worth noting that the cross product, in the sense that we use it in $\Bbb R^3$, doesn't exist in $\Bbb R^n$, generally (it does exist when $n = 7$, but that is the only other case).

There is a generalization of the cross product called the wedge product, but we use that in different ways. The cross product is something that we defined because we happen to live in a world of three spatial dimensions, and it allows us to compactify our notation of how we handle vector operations in this world.

When you want to investigate similar operations on vector spaces of other dimensions, you lose this convenience, and have to use the wedge product instead. So the cross product is something that we defined because it happens to be very handy and useful in real-world applications.