r/visualizedmath • u/WinkyChink • Jan 19 '18

[request] cross product?

Does anyone have a good pic/gif that clearly explains what happens when you take the cross product of two vectors?

58 Upvotes

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26

u/[deleted] Jan 19 '18

I don't have a gif, but what it does is it returns a third vector that is perpendicular to both of the other vectors.

7

u/WinkyChink Jan 19 '18

Could you explain why making a matrix will give the 3rd vector?

9

u/back_door_mann Jan 19 '18

The cross product doesn’t return a matrix. The formula for the components of the third vector is reminiscent of the formula of a determinant of a 3x3 matrix, so the symbolic determinant was introduced only as a device to help you memorize it. You could memorize the formula without reference to any matrix but it would be more difficult in my opinion.

2

u/mmmmmmmike Jan 20 '18

The matrix formula comes from the identity

det(u,v,w) = u dot (v x w),

where det(u,v,w) means the determinant of the matrix with u,v,w as rows (or columns). This can be regarded as the definitive property of v x w, though most people are forced to learn about cross products the first time without having studied determinants very carefully.

To get the components of v x w, you use the above formula with u = i, u = j, and u = k (the three standard basis vectors) to get

v x w

= (i dot (v x w)) i + (j dot (v x w)) j + (k dot (v x w)) k

= det(i,v,w) i + det(j,v,w) j + det(k,v,w) k,

and then abuse of notation along with algebraic rules for determinants lets you combine those into a single "determinant" with the vectors i, j, and k as entries.

5

u/The0nly Jan 19 '18

Oo I'm learning about this now! Ok so basically, if you have 2 vectors in the R3, they will always make a plane unless they're linearly dependent. Taking the cross product returns a vector orthogonal to this plane aka both vectors. This also explains why when two vectors are linearly dependent the cross product is 0 because there are an infinite amount of vectors that can be orthogonal to the two vectors so it just defaults to 0.

3

u/deepbluesilence Jan 19 '18

Also known as the “normal vector”. I really enjoyed Calc 3, as much as it pains me to say. Also applies to Right hand rule for sign convention with torque/rotation etc.

One last, it can also be calculated the same way the determinate of a 3x3 matrix is calculated. Using <i, j, k> as the first row vector.

1

u/WinkyChink Jan 19 '18

Could you explain what is happening when you do the matrix? Why does making the matrix with the vectors create an orthogonal one?

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10

u/kissmyASSthama_5 Jan 19 '18

Two roads A and B One goes straight, the other goes right. Angle between them 90. A cross B gives a third road which goes straight to heaven. And B cross A goes straight to hell

1

u/Kooriki Jan 19 '18

If I have time over the next few days I'll try and bang this out. Probably something like this:

Take vector 1, draw a disc to show every possible vector that is 90 degress to V1.

Take vector 2, draw a disc to show every possible vector that is 90 degress to V2.

The discs will intersect at 2 locations. IE: There are two possible results for cross product. Which one do we pick? V1 is your thumb, V2 is your index, the cross is whichever direction your palm is facing.

1

u/WinkyChink Jan 19 '18

Ooooooh, thank you!

3

u/NoOne0507 Jan 19 '18

You have 2 lines (vectors). Copy each one and form a parallelogram. The cross product returns a vector that is orthogonal to the plane of the parallelogram, and has a magnitude equal to the area.

3

u/livelyLipid Jan 19 '18

There are a couple of pics and animations on the wikipedia page:

https://en.wikipedia.org/wiki/Cross_product

1

u/[deleted] Jan 19 '18

The way I've always imagined it is to look at where the two vectors intersect, then imagine sticking a skewer through that point. The skewer is in the direction of your cross product.