Vector Rotation About Arbitrary Axis

[MD Aminuzzaman]

I am new to this forum. I was reading this document :
http://math.kennesaw.edu/~plaval/math4490/rotgen.pdf

Here the author says that from this figure
http://i.stack.imgur.com/KBw9l.png

that we can express $v_{\perp}$ like this :
$$T (v_{\perp}) = \cos(\theta) v_{\perp} + \sin(\theta) w$$

I don't understand this part. Can anybody explain how $$T(v_{\perp}) $$ is $$\cos(\theta) v_{\perp} + \sin(\theta) w$$?

 

[MD Aminuzzaman]

Any Help?

 

[Erland]

Isn't this obvious from the figure? I see no problem at all.

Although what should be a circle looks like an ellipse in the figure I see on my computer. Could be something wrong with the graphics the author used. Anyway, if you also see an ellipse, think of it as a circle!

 

[MD Aminuzzaman]

Why we are experessing the Vector as addition of cos(θ)v⊥ and sin(θ)w ? Is it the result of 2 vector addition like Vab = Va + Vb?

 

[Erland]

Is your question why it is correct to do this, or what is the use of doing this?

 

[MD Aminuzzaman]

why it is correct to do this ?

 

[Erland]

Don't you know how to add vectors geometrically, with the parallellgram law?

Here we are studying a linear tranformation which rotates every vector in the plane perpendicular to the rotation axis by the angle $\theta$ about that axis. Then $v_\perp$ is mapped to $T(v_\perp)$ in the figure. And $v_\perp$ and $w$ are perpendicular unit vectors in this plane, so we obtain the relation stated.

 

[MD Aminuzzaman]

Thanks now i understood. Sorry sometimes i am dumb and gets confused with simpler things.

 

[Erland]

Thanks now i understood. Sorry sometimes i am dumb and gets confused with simpler things.

No problem, you're very welcome! We all get confused sometimes.