Skip to main content

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 \( \cos \phi\).
Figure 1. Two projections of an unit vector \(\mathbf{h}\).

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{eqnarray*}
 |\mathbf{h}_1| &=&  \mathbf{h}\cdot\mathbf{e}_1\\
 &=& |\mathbf{h}| |\mathbf{e}_1| \cos \alpha \\
 &=& \cos \alpha.
\end{eqnarray*}
If you noticed \(\mathbf{h}_1\) is a projection of \(\mathbf{h}'\) on \(\mathbf{e}_1\) axis,
\begin{eqnarray*}
 \mathbf{h}'\cdot\mathbf{e}_1 &=& |\mathbf{h}'||\mathbf{e}_1| \cos \phi \\
 &=& |\mathbf{h}'| \cos \phi.
\end{eqnarray*}
This is actually \(|\mathbf{h}_1|\). See the Figure 2. Therefore,
\begin{eqnarray*}
 \cos \alpha &=& |\mathbf{h}'| \cos \phi.
\end{eqnarray*}
Figure 2. The relationship among \(\mathbf{h} \), \(\mathbf{h}'\) and \(\mathbf{h}_1\).
In the end, if you noticed \(\mathbf{h}_1\) is a direct projection of \(\mathbf{h}\) or projection of projection, there is no mystery here.

When I understand this problem as above, I recall directional cosine.  I learned this long time ago, but this idea is located nowhere in my head and no connection with other mathematical ideas. I just remember the squared sum of the cosines becomes 1. It's a kind of left out idea for me. But, if the projection is on the each axis, this is directional cosine. Then the squared sum is of course one. (Or the square root of squared sum is one. But square root of one is anyway one.) Because this is the length of unit vector, which is one. It's so simple. I don't know why I never realize the directional cosine is just coordinate elements of unit vector.

Labels:

Comments

Post a Comment

Popular posts from this blog

Gauss's quote for positive, negative, and imaginary number

Recently I watched the following great videos about imaginary numbers by Welch Labs. https://youtu.be/T647CGsuOVU?list=PLiaHhY2iBX9g6KIvZ_703G3KJXapKkNaF I like this article about naming of math by Kalid Azad. https://betterexplained.com/articles/learning-tip-idea-name/ Both articles mentioned about Gauss, who suggested to use other names of positive, negative, and imaginary numbers. Gauss wrote these names are wrong and that is one of the reason people didn't get why negative times negative is positive, or, pure positive imaginary times pure positive imaginary is negative real number. I made a few videos about explaining why -1 * -1 = +1, too. Explanation: why -1 * -1 = +1 by pattern https://youtu.be/uD7JRdAzKP8 Explanation: why -1 * -1 = +1 by climbing a mountain https://youtu.be/uD7JRdAzKP8 But actually Gauss's insight is much powerful. The original is in the Gauß, Werke, Bd. 2, S. 178 . Hätte man +1, -1, √-1) nicht positiv, negative, imaginäre (oder gar um...
Post a Comment
Read more

Tezuka Osamu's Black Jack, "Shrinking"

I like several novel authors. My first favorite author is probably Teduka, Osamu. I still love him. The list grows by adding Hoshi, Shinichi, Agatha Christie, Hermann Hesse, and so forth. My first favorite article of Tezuka was Atom as most of the (boy's) Tezuka fans did. But my favorite is Black Jack. I try to summarize one story, it is still quite vivid in my memory. I first read this story when I was 13 - 15 years old. I re-read it at least several times since Black Jack is composed of many short episodes. The title should be "ちぢむ (SHRINKING)" or it might be "縮む(Shrinking)". (It is not so convenient to translate this to English, since English does not have a system to say the exact same word in several ways. So I just simulate it with capital letters.) Black Jack is a genius surgeon, but he does not have the license. In short, his medical activity is illegal. His skill is top level in the world, but, the fee is also out-of-law expensive. In the story ...
Post a Comment
Read more

Why A^{T}A is invertible? (2) Linear Algebra

Why A^{T}A has the inverse Let me explain why A^{T}A has the inverse, if the columns of A are independent. First, if a matrix is n by n, and all the columns are independent, then this is a square full rank matrix. Therefore, there is the inverse. So, the problem is when A is a m by n, rectangle matrix.  Strang's explanation is based on null space. Null space and column space are the fundamental of the linear algebra. This explanation is simple and clear. However, when I was a University student, I did not recall the explanation of the null space in my linear algebra class. Maybe I was careless. I regret that... Explanation based on null space This explanation is based on Strang's book. Column space and null space are the main characters. Let's start with this explanation. Assume  x  where x is in the null space of A .  The matrices ( A^{T} A ) and A share the null space as the following: This means, if x is in the null space of A , x is also in the n...
Post a Comment
Read more