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Showing posts from July, 2012

Vector projection and directional cosine

Abstract When I compute something, I see it is correct, but sometimes my intuition doesn't work. Especially, I am not good at statistics, though, sometimes I don't see geometry also. I can show you such example this time.

Vector projection and directional cosine My friend Dietger asked me a problem. Figure 1 shows the problem. Let \(\mathbf{h}'\) is a projected vector of \(\mathbf{h}\) on the \(\mathbf{e}_1,\mathbf{e}_2\) plane of an orthogonal coordinate system. Where \(\mathbf{h}\) is an arbitrary unit vector. Let \(\mathbf{h}_1\) is the projection of \(\mathbf{h}\) on the axis \(\mathbf{e}_1\). Then show
\[
 \cos \alpha = |\mathbf{h}'| \cos \phi.
\]
It's intuitively odd for me that there is a length ratio \(h'\) between \(\cos \alpha\) and \( \cos \phi\).

However, these are projections, therefore some \(\cos\)  relationships. Let's start with  \(\mathbf{h}_1\).  \(\mathbf{h}_1\) is a projection of  \(\mathbf{h}\) on \(\mathbf{e}_1\) axis,
\begin{eqnarra…

MathJax Test

It's bit cumbersome to create an image for each equation every time. So I try MathJax here. This is a great software!

To enable MathJax, Add
<script type="text/javascript" src="http://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML"> </script> in the blog text in blogger. Now some equation here.

\[
 |\mathbf{h}_1| =  \mathbf{h}\cdot \mathbf{e}_1
 = |\mathbf{h}| |\mathbf{e}_1| \cos \alpha
 = \cos \alpha
\]
becomes \[  |\mathbf{h}_1| =  \mathbf{h}\cdot\mathbf{e}_1  = |\mathbf{h}| |\mathbf{e}_1| \cos \alpha  = \cos \alpha \] Awesome.