# Thread: Transformation matrix, vector algebra word problem

1. ## Transformation matrix, vector algebra word problem

Hi everyone. I am not sure if this problem belongs under the "advanced algebra" section but it seemed like it may. Please let me know if there is a different section that would better fit this problem.

So here is a word problem that is proposed:
A solar panel is capable of rotating independently about a fixed x,y,z axis. The solar panel start by pointing upward in the z direction and the center of the pannel is located at (0,0,0). At a certain time of day the sun relative to the x y z origin is in the direction of the vector (1,1,1). Calculate the transformation matrix which can be applied to the original solar panel such that the maximum power is obtained (normal to panel is aligned with sun direction). Note that the center is kept at (0,0,0).

If anyone has an idea how to set up and solve this problem please post. Thank you for your help!

2. Look attachment.
When more sun light pass through light section, solar panel will get more power.
So we need get maximum $\displaystyle projection\ of\ \vec{P_2P_1}\ on\ light\ section\ \vec{u}$

$\displaystyle Power=Proj_{\vec{u}}\vec{P_2P_1}=(\vec{u}\cdot \vec{P_2P_1})\vec{u}= (\vec{u}\cdot (\vec{P}_1- \vec{P}_2))\vec{u}$$\displaystyle = (\vec{u}\cdot (2\vec{P}_1))\vec{u}=2(\vec{u}\cdot \vec{P}_1)\vec{u}$
$\displaystyle \|Power\|=\|2(\vec{u}\cdot \vec{P}_1)\vec{u}\|=2\|\vec{u}\cdot \vec{P}_1\|\cdot \|\vec{u}\|,\ \because \|\vec{u}\|=1$
$\displaystyle \therefore\|Power\|=2\|\vec{u}\cdot \vec{P}_1\|=2\|\vec{u}\|\cdot \|\vec{P}_1\|\cdot \|cos(\theta)\|=2\|\vec{P}_1\|\cdot \|cos(\theta)\|$
When $\displaystyle \theta=0$, it means $\displaystyle \vec{P_2P_1}\ parallel \ light\ section\ \vec{u}$, solar panel get maximum power.
In 3 dimension coordination, we set plane of solar panel orthogonal to Sun light.