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# Vectorized Solution to Rotate Multiple Points each at a Different Angle

Asked by John deGrassie on 7 Mar 2013

I want to rotate a matrix of points, i.e. column vectors. However, I want to rotate each vector in the matrix by a different angle. For example:

```pointMatrix  = [v1,v2,v3,v4]; %vi is a column vector
rotateAngles = [10,20,30,40]; %degrees
```

Say I want to rotate these points around the z-axis. Therefore, for one point I could do something like the following:

```Rz = [[cos(rotateAngles(1)) -sin(rotateAngles(1)) 0];...
[sin(rotateAngles(1))  cos(rotateAngles(1)) 0];...
[       0                      0            1]];
v1Rotated = Rz*v1;
```

Is there a non-loop way to rotate all the vectors in my pointMatrix by each one's unique rotation angle? Something like this...

`   allRotatedPoints = superRotationMatrix*pointMatrix;`

where the superRotationMatrix "magically" rotates each column by the corresponding angle.

Thanks!

## Products

Answer by Teja Muppirala on 8 Mar 2013
Edited by Teja Muppirala on 8 Mar 2013

This vectorized solution uses complex exponentials and works about 2 orders of magnitude faster for large vectors.

```M = exp(rotateAngles*1i) .* ([1 1i 0]*pointMatrix);
allRotatedPoints = [real(M); imag(M); pointMatrix(3,:)];
```

Answer by Matt J on 7 Mar 2013
Edited by Matt J on 7 Mar 2013

You can use MTIMESX on the file exchange

http://www.mathworks.com/matlabcentral/fileexchange/25977-mtimesx-fast-matrix-multiply-with-multi-dimensional-support

If your superRotationMatrix is 3x3xN and you reshape your pointMatrix to be 3x1xN, then

```mtimesx(superRotationMatrix,pointMatrix)
```

will give you the rotated vectors in a 3x1xN output.