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?

The first term: \cos(\theta) v_{\perp} should be evident from the figure. The second term is a bit trickier to understand - the w vector has magnitude equal to v_{\perp}, but is at 90 degrees to it. So the magnitude of the ( \sin \theta) w term is equal to the magnitude of ( \sin \theta) v_{\perp}, but points in the w direction. The vector w in turn is defined by \hat r \times \bold v . Hope this helps.

Hello, Thanks for the answer. I understood that part. But thing i did not understood is that how can we express v_{\perp} as addition of \cos(\theta) v_{\perp} + \sin(\theta) w , that is the addition of its 2 component \cos(\theta) v_{\perp} and \sin(\theta) w. Is it same as meaning of 2 vector addition like Vab = Va + Vb?