# Questions tagged [vectors]

Use this tag for questions and problems involving vectors, e.g., in an Euclidean plane or space. More abstract questions, might better be tagged vector-spaces, linear-algebra, etc.

2,059 questions
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### What is the difference between tensor calculus and exterior derivative type concepts?

I am trying to clarify terms in order to help me figure out what I'd like to study. I understand that $p$-forms and $p$-vectors are used with things like wedge products, exterior algebras, and a ...
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### Bipolar toroidal coordinates - position vector, velocity and acceleration

Bipolar toroidal coordinates: $x = a \frac{\sinh\tau \cos\phi}{\cosh\tau-\cos\sigma}$ $y = a \frac{\sinh\tau \sin\phi}{\cosh\tau-\cos\sigma}$ $z=a \frac{\sin\sigma}{\cosh\tau-\cos\sigma}$ Would ...
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### Finding a summarizing vector for average angle calculation

Let $L$ and $R$ be two bags of positive vectors such that all vectors have length $k$. Define the distance $d_{avg}$ between the bags as the average pairwise angle between the vectors. Is is possible ...
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### Is this the correct solution involving vector subspaces and basis?

I need to find the basis and hence dimension of a subspace of $\mathbb{R^3}$. 1) $$U=\{(x,y,z):x=2y\}$$ Solution: We have $x=2y \iff y=\frac{x}{2}$ therefore we can write all elements in $U$ as the ...
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### Understanding the Jacobian past calculus

What's taught in calculus: In the calculus of multiple variables I learned that the Jacobian \textbf J=\frac{\partial(x_1,\ldots,x_n)}{\partial(t_1,\ldots,t_n)}=\left(\begin{array}{ccc}\frac{\...
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### Add vectors from a set to reach the goal vector, using the minimum possible cost

I am trying to solve a problem in an optimal way. The problem is as follows: We have an n-dimensional space In this space, we have a "finish" point with n coordinates, all non-negative We have a set ...
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### Cross check for the derivative of a unit vector $\frac{x}{|x|}$

Can you please help me in finding out the mistakes I am doing during the calculation of derivative of a vector. I am briefing the problem I am trying to solve as follows. There is a line joining two ...
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### Given an arbitrary number of vectors, how can I find a vector with the average angle without using trig functions

I have $n$ unit vectors vectors $v_1...v_n$ in $\mathbb{R}^2$, each with respective angles $\theta_1...\theta_n$, which corresponds to $tan(\frac{y}{x})$. I want to find a vector with the average ...
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### Prove a vector calculus identity

Let $V \subseteq \mathbb{R}^3$ have smooth boundary $\partial V$. Suppose $f, g, \mathbf{J}$ are differentiable on $V \cup \partial V$, with $\nabla \cdot \mathbf{J} = 0$ in $V \cup \partial V$, then ...
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### Relation between curl of vector field and and its derivative in the normal direction

I have been trying to prove this equivalence but I could not manage to do it. I have used all kinds of identities with no success. I want to prove that, being $\mathbf n$ a unit surface normal ...
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### Why is it important to get the cosine of an angle theta instead of the angle itself in two unit vectors?

I'm a bit confused about why is it important to get the cosine of an angle theta instead of the angle itself in two unit vectors? I mean, to get the cousine theta angle we need to dot product of the ...
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### What is the maximal number of such vectors?

$n,k$ are integers and I need to find the maximal integer $M$ such that there exist $M$ different n-dimensional vectors whose components are in $\{-1,1\}$ satisfying: any two of $M$ vectors has ...