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…

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…