Posts

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!

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