@Asterix For one thing, it gives a meaning to curl in 2D (which is useful for all sorts of things, like 2D or "thin" fluid equations). It's also a useful shorthand for talking about areas in 2D. I.e., if I have a triangle with vertices $a$, $b$, and $c$ in the plane, I can write its (signed) area as

$$\tfrac{1}{2}\left((b-a) \times (c-a)\right).$$

zyx

Does the definition of cross product depend on the fact that we use a right-handed coordinate system? (As in, the basis (e1, e2, e3) are such that e1 x e2 = e3) If this is true, wouldn't we need to double check the coordinate system before using the rule?

PS: Also found this interesting read on 'pseudo-vectors' and how they're special: https://www.quora.com/What-is-a-pseudovector/answer/Brian-Bi

dvanmali

This slide confused me in class because the cross product (u x v) helps to compute the axis of rotation about vectors u and v. The diagram seems to originate on the center of the area for vector u and vector v which doesn't seem correct in vector space. Please correct me though if I am wrong! :D

jkalapos

Vectors are just magnitude and direction so there is no requirement for the cross product vector (u x v) to originate from the intersection of u and v. Even the intersection of u and v isn't a thing because they're also both just vectors. As long as its pointing in the right direction with the right norm (u x v) can be anywhere

keenan

@dvanmali Yeak, @jkalapos is totally right: when we talk about vectors in a vector space, they have no "basepoint"; this idea is mentioned briefly at the bottom of this slide. When drawing pictures, this means we can draw little arrows in more suggestive places; here the arrow for u x v suggests that is the normal of a little planar region in space.

What application may use cross product in 2D?

@Asterix For one thing, it gives a meaning to curl in 2D (which is useful for all sorts of things, like 2D or "thin" fluid equations). It's also a useful shorthand for talking about areas in 2D. I.e., if I have a triangle with vertices $a$, $b$, and $c$ in the plane, I can write its (signed) area as

$$\tfrac{1}{2}\left((b-a) \times (c-a)\right).$$

Does the definition of cross product depend on the fact that we use a right-handed coordinate system? (As in, the basis (e1, e2, e3) are such that e1 x e2 = e3) If this is true, wouldn't we need to double check the coordinate system before using the rule?

PS: Also found this interesting read on 'pseudo-vectors' and how they're special: https://www.quora.com/What-is-a-pseudovector/answer/Brian-Bi

This slide confused me in class because the cross product (u x v) helps to compute the axis of rotation about vectors u and v. The diagram seems to originate on the center of the area for vector u and vector v which doesn't seem correct in vector space. Please correct me though if I am wrong! :D

Vectors are just magnitude and direction so there is no requirement for the cross product vector (u x v) to originate from the intersection of u and v. Even the intersection of u and v isn't a thing because they're also both just vectors. As long as its pointing in the right direction with the right norm (u x v) can be anywhere

@dvanmali Yeak, @jkalapos is totally right: when we talk about vectors in a vector space, they have no "basepoint"; this idea is mentioned briefly at the bottom of this slide. When drawing pictures, this means we can draw little arrows in more suggestive places; here the arrow for u x v suggests that is the normal of a little planar region in space.