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rmi111 · Dec 11, 2015, 07:17 AM

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 $\cos(\theta) v_{\perp} + \sin(\theta) w$ ?

ebaines · Dec 11, 2015, 09:50 AM

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.

rmi111 · Dec 11, 2015, 11:37 AM

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?