Consider the following 3 vectors:
$$\begin{align} \vec{TL} = \begin{bmatrix} a \\ b \\ c \end{bmatrix} \\ \vec{TR}=\begin{bmatrix} d \\ e \\ f \end{bmatrix} \\ \vec{BL} = \begin{bmatrix} g \\ h \\ \alpha \end{bmatrix} \end{align} $$Where $a,b,c,d,e,f,g,h \in \mathbb{R}$, and are 'known' scalars, and $\alpha \in \mathbb{R}$, and is an 'unknown' scalar.
The vectors $\vec{TL},\vec{TR}, \vec{BL}$ are the 3 respective corners1 of a square grid.
Assume the necessary conditions for the above to be true, that is, the minimum distance between $\vec{TL}$ and $\vec{TR}$ is less than or equal to the distance between $\vec{TL}$ and $\vec{BL}$.
Find an expression for $\alpha$ in terms of $a,b,c,d,e,f,g,h$.
Suppose we have found a value for $\alpha$ where the previous conditions are satisfied, so let $\alpha$ be the scalar $i$. Hence, the vectors $\vec{TL},\vec{TR}, \vec{BL}$ describe a square grid, of dimensions $N\times N$, where $N\in \mathbb{R^{+}}$. There exists a highlighted cell on this grid, on row $R$ and column $C$, where $R,C \in \mathbb{R^{+}};R,C\leq N$.
State a vector, $\vec{A}$, which intercepts the centre of this highlighted cell from the origin, in terms of $a,b,c,d,e,f,g,R,C,N$.
Given that $d \geq g$, $e\geq h$, $f\geq i$, determine the upper and lower bounds for $x,y,z$ if:
$$\begin{align} \vec{A}=\begin{bmatrix} x \\ y \\ z \end{bmatrix} \end{align} $$is a vector which passes through any point inside the highlighted cell from above, from the origin.
Given that $d \leq g$, $e\leq h$, $f\leq i$, determine the upper and lower bounds for $x,y,z$ if:
$$\begin{align} \vec{A}=\begin{bmatrix} x \\ y \\ z \end{bmatrix} \end{align} $$is a vector which passes through the highlighted cell from above, from the origin.
i.e. top left corner, top right corner, bottom left corner, respectively. ↩
