Prove to myself that rotation is a linear transformation?

[davidbenari]

How do you prove that rotation of a vector is a linear transformation?

It's intuitive (although not completely crystal clear to me) that it is a linear transformation at the 2d level, but how do I prove it to myself (that this is a general property of rotations)?

For example, rotate vector $\vec{V}$ in the xy plane by 30º. $Rot(\vec{V})=Rot(Vx \vec{e1} + Vy \vec{e2} + Vz \vec{e3})= Vx Rot(\vec{e1}) +Vy Rot(\vec{e2}) + Vz Rot(\vec{e3})$

Or in other words, if I have to rotate a vector on the plane xy (for example), how do I prove that this rotation can be done by rotating only my base vectors (or my axis, if you will) and then drawing my original vector with that "new" base or coordinate axis.

Thanks.

 

[Stephen Tashi]

What is your definition of a rotation?

To prove something, you'd have to begin with a precise definition of a rotation. If you take an intuitive idea of a rotation as your starting point, you can't do anything except make an intuitive argument.

 

[davidbenari]

I think what my class means by rotating a vector is the following: Draw a coordinate axis. Draw a position vector in the traditional way (as an arrow). If I say rotate a vector 30º in the xy plane then this means to trail an arclength equivalent to 30º with the head of your arrow keeping the tail fixed on the origin.

Is this a good conception of what it means to rotate a vector in 3d?

If it's worth mentioning, this is a mathematical physics class, so maybe you have a more precise idea of what they mean by "vector rotation".

Thanks.

 

[davidbenari]

What's the rigorous definition, by the way?

 

[Stephen Tashi]

One way to state a rigorous definition of a 2-D counterclockwise rotation by angle theta about the origin (0,0) is to give the equations that specify the image of point (x,y). The rotation is a function R(x,y,theta) gives a 2-D vector R(x,y,theta) = ( f(x,y, theta), g(x,y,theta) ). What functions would f and g be?

If you know about matrices, look at the Wikipedia article on rotation matrix: http://en.wikipedia.org/wiki/Rotation_matrix That will tell you the formulas for f and g.