Align two Arbitrary 2D vectors

[Peter5897]

I've got a computer graphics final coming up and for some reason my brain is crapping out when it comes to, what should be, a simple problem.

I know how to align two arbitrary 3D vectors (axis-angle) but 2D in 2D I'm having trouble remembering how to do it.

On my review sheet it states

Homework Statement

In 2D, computer the 3x3 matrix that aligns V = (Vx, Vy) with W = (Wx, Wy)

Homework Equations

I suppose the rotation matrix would be a start.

----------------------------
| cos(theta) -sin(theta) 0 |
| sin(theta) cos(theta) 0 |
| 0 0 1 |
---------------------------

The Attempt at a Solution

I know that theta = arccos(v dot w) but I believe that on a test shoving in an arccos won't be sufficient. IE

--------------------------------------------------
| cos(arccos(v dot w)) -sin(arccos(v dot w)) 0 |
| sin(arccos(v dot w)) cos(arccos(v dot w)) 0 |
| 0 0 1 |
---------------------------------------------------

Pretty gross looking if you ask me.

My question to you is, what am I forgetting? I believe there must be some simple linear algebra theorem that I am currently drawing a blank on, or is this really the best way?

Thanks for any insight you guys could bring,
-Peter

PS, sorry for my awful attempt at making matrices, the formatting seems to change during the submission process.

PPS, probably should have put this in the pre-calc section... whoops.

 

[lippyka]

The answer is to use the cross product of the two vectors to find the angle between them. You can then use the rotation matrix with that angle to align the two vectors.So, in this case, you'd calculate the cross product as follows:V x W = (Vx * Wy - Vy * Wx, Vy * Wx - Vx * Wy)Then use that result to find the angle between them using:theta = arctan2(V x W)Finally, use the theta in the rotation matrix to find the 3x3 matrix that will align V and W:----------------------------| cos(theta) -sin(theta) 0 || sin(theta) cos(theta) 0 || 0 0 1 |---------------------------