Articles of vectors

Linear dependence in Carathéodory's theorem (convex hull)

I don’t get this step in proof of Carathéodory’s theorem (convex hull) Why: Suppose k > d + 1 (otherwise, there is nothing to prove). Then, the points $x_2 − x_1, …, x_k − x_1$ are linearly dependent Why is this true? How can we cay these points are linearly dependent?

How to find the point on the sphere that is closest to a plane?

Consider the plane $x+2y+2z=4$, how to find the point on the sphere $x^2+y^2+z^2=1$ that is closest to the plane? I could find the distance from the plane to the origin using the formula $D=\frac{|1\cdot 0+2\cdot 0+2\cdot 0-4|}{\sqrt{1^2+2^2+2^2}}=\frac43$, and then I can find the distance between the plane and sphere by subtracting the radius of sphere […]

Calculating a “max support flux vector sequence”

Tl;dr: what is a max support flux vector sequence? In a paper (arXiv link; full citation below) on continuous chemical reaction networks (CCRNs)$^1$ it discusses an algorithm for reaching different states using various chemical reactions. I have some questions about a line of the algorithm and what it means given the authors’ notation. Preliminaries The […]

Geometry in Vectors

Let ${A} = \begin{pmatrix} \cos \frac{2 \pi}{5} & -\sin \frac{2 \pi}{5} \\ \sin \frac{2 \pi}{5} & \cos \frac{2 \pi}{5} \end{pmatrix}$ and ${B} = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$. Let $S$ be the set of all matrices that can be generated from taking products of ${A}$ and ${B}$, in any order. For […]

Prove that a particle is traveling on a plane from its velocity and acceleration in space

We have a particle that travels in 3d space. It is given that $a(t)=-r(t)$. First I need to prove that: $$\frac{d}{dt}(\underline r \times \underline v)=0$$ This is easily done: $$ \frac{d}{dt}(\underline r \times \underline v)=r(t) \times r”(t)+r'(t) \times r'(t)=0 $$ Then I need to prove that the vector $\underline r \times \underline v$ is a […]

How do I determine if 3 vectors are collinear?

I only know how to show that 2 vectors are collinear, but for 3 vectors I only know how to prove coplanarity.

Problems with the definition of vectors as directed line segments in $\mathbb{R}^3$

First I’ll say where I’m working: The vectorial spaces $\mathbb{R}^2$ and $\mathbb{R}^3$. Then I’ll define a vector of this spaces as the following: $\textbf{Definition. }$ A vector $\vec{v}$ is the set of all equal directed line segments. Now suppose that $$\underbrace{\overrightarrow{AB}}_{\mbox{directed line segment}} \in \vec{v},$$ which is a correct notation, by definition. So why do […]

Why do vectors and points use similar notations?

Alright a vector is suppose to be magnitude with direction or rather in a simple sense, “numbers with direction”. What really gets to me is that why would a point and a vector share similar notation for describing themselves ? Lets take a point in a 3D space $(2,3,4)$, and then lets take a vector […]

How to convert components into an angle directly (for vectors)?

Let us say we have a vector with $x$-component $-2$, and $y$-component $-1$. We have the equation: $$\tan\theta=\frac{-1}{-2}$$ So if we take the inverse of $\tan$ of $\frac12$ we get $26.565^\circ$. The problem is this is wrong, it’s $206.565^\circ$. The problem is apparent; the equation above has multiple solutions. The inverse of $\tan$ occasionally gives […]

Shortest distance between two moving points

So I found this question on the Internet, which turned out more tricky than I thought: ” The position of boat A is given by $x(t)=3-t$ and $y(t)=2t-4$ The position of boat B is $x(t)=4-3t)$ and $y(t)=3-2t)$ respectively. Find the value of $t$ for which the boats are closes to each other. (distances are in […]