I am trying to understand two parts from the picture below in my textbook, but I dont understand how they arrived at it.
I am trying to understand the proof below and how they got $P_L(\vec{v}) = \vec{v}+\frac{1}{2}[Q_L(\vec{v})+\vec{v}]$, as well as how they got the $3 \times 3$ matrix for the projection $P_L(\vec{v})$ from the second step.
Please help!
Thanks
The first equation comes directly from geometric representation of vector addition and subtraction:
$$P_L(\vec{v}) = \vec{v}+\frac{1}{2}[Q_L(\vec{v})-\vec{v}]$$
As you see from the picture, $Q_L(\vec{v})-\vec{v}$ is the vector pointing from the end of $\vec{v}$ to the end of $Q_L(\vec{v})$. The $\frac{1}{2}$ make it half of the length. Now when you add $\vec{v}$ and $\frac{1}{2}[Q_L(\vec{v})-\vec{v}]$, you connect from the end of $\vec{v}$ this half length vector, you will get $\vec{v}+\frac{1}{2}[Q_L(\vec{v})-\vec{v}]=P_L(\vec{v})$.
For the matrix, if you understand the step
$$\frac{ax+by+cz}{a^2+b^2+c^2}\begin{bmatrix}a\\b\\c\end{bmatrix}$$
The next step just needs a verification from matrix multiplication from right hand side to left hand side.