# Check if a point lies beneath vectors convex to it?

I am trying to check if a point lies beneath vectors convex to it by trying to deduce which of the four cases below fits any given scenario:

I am hoping to isolate B, in that there's some general way to deduce whether a given scenario is case B or not (without the need to compute angles between vectors) i.e. since only $$\vec{a}$$ and $$\vec{b}$$ in case B are convex to the point $$p$$

Just find the coordinates of $$p - h$$ with respect to the frame whose axes are vectors $$\vec{a}, \vec{b}$$ and whose origin is $$h$$

The associated linear system can be written as

$$[ \vec{a} , \vec{b} ] [x_1, y_1]^T = p - h$$

Then, it follows that

$$[x_1, y_1]^T = [ \vec{a}, \vec{b} ]^{-1} (p - h)$$

Since you're interested in Case $$B$$, then you want $$x_1 \gt 0 , y_1 \gt 0$$.

For example, if point $$h = [2, 3]^T , \vec{a} = [1, -3]^T , \vec{b} = [2, 6]^T$$, and $$p$$ is given as $$[ 5, 7 ]^T$$, then

$$\begin{bmatrix} x_1 \\ y_1 \end{bmatrix} = \begin{bmatrix} 1 && 2 \\ -3 && 6 \end{bmatrix}^{-1} \begin{bmatrix} 3 \\ 4 \end{bmatrix}$$

And this solves to

$$\begin{bmatrix} x_1 \\ y_1 \end{bmatrix} = \dfrac{1}{12} \begin{bmatrix} 6 && -2 \\ 3 && 1 \end{bmatrix} \begin{bmatrix} 3 \\ 4 \end{bmatrix} =\dfrac{1}{12} \begin{bmatrix} 10 \\ 13 \end{bmatrix}$$

Hence, $$x_1 \gt 0 , y_1 \gt 0$$, so this is Case $$B$$.

• Hi @Robin’sPremiumCoffee would this still work if $\vec{a}$ and $\vec{b}$ for example are not perpendicular to one another, also say we are dealing with a 3D frame $\vec{a} , \vec{b} ,\vec{d}$ instead would it be okay to apply the same logic asper: $[x_1, y_1, z_1]^T = [ \vec{a} , \vec{b} ,\vec{d}]^{-1} (p - h)$ such that $x_1 \gt 0 , y_1 \gt 0, z_1 \gt 0$ May 8, 2022 at 14:19
• Yes. It also works if $\vec{a}, \vec{b}$ are not mutually perpendicular. And the equation you wrote for $3D$ is also correct, and its conclusion is correct. May 8, 2022 at 15:05
• Hi @Hosam Y. Hajjir, please could you help out with this math.stackexchange.com/q/4484719/585488 Jul 3, 2022 at 7:57
• What do you mean by polytope ? What does that mean ? Jul 3, 2022 at 7:59
• just a polytope in general e.g. a 2-polytope (polygon), 3-polytope (polyhedron/3d mesh) .etc you may also have a look at en.m.wikipedia.org/wiki/Polytope Jul 3, 2022 at 8:06