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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.

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1k views

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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77 views

Is there an accepted terminology/notation for the vector $\langle \theta \rangle := \begin{bmatrix} \sin \theta \\ \cos \theta \end{bmatrix}$?

Suppose you start off at a point $A$, walk 10 units at a bearing of $\frac{\pi}{4}$, then walk 25 more at a bearing of $\frac{5\pi}{4}$. The point where you end up is, of course, $$A+10\left\langle \...
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804 views

What does it mean for a polar coordinate system to have basis vectors?

So I understand that every element of a vector space can be represented uniquely by a linear combination of the basis vectors: $v=\alpha_1v_1+\cdots+\alpha_nv_n$ Then coordinates to those basis ...
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200 views

How to show that this function respects the strict ordering of its input.

Suppose you have a vector $\pmb x=\{x_i\}_{i=1}^n$ where each entry is drawn from a continuous distribution and $n$ is even. Then, denote $i^*=\{1\leq j\neq i\leq n:|x_i-x_{j}|=\mbox{med}_j|x_i-x_j|\}...
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4k views

How to find the “average” direction of a set of vectors?

I have a series of vector directions and I need to find the "average" direction. I am not looking for the overall direction which would be the sum of the directions and this can't be used in cases ...
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28 views

Question on elementary Linear Algebra product

I came across this in a problem: $$\frac{\mathbf{u}\mathbf{v}^{T}\mathbf{A}^{-1}+\mathbf{u}\mathbf{v}^{T}\mathbf{A}^{-1}\mathbf{u}\mathbf{v}^{T}\mathbf{A}^{-1}}{1+\mathbf{v}^{T}\mathbf{A}^{-1}\mathbf{...
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179 views

Comparing elements of sets

Let $a_1, a_2, a_3, a_4$ be real numbers. Consider the following sets $$ \mathcal{U}_1\equiv \{-a_1, a_1, -a_2, a_2, 0, \infty, -\infty\} $$ $$ \mathcal{U}_2\equiv \{-a_3, a_3, -a_4, a_4, 0, \infty,...
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78 views

A term for a 'vector' that only has an orientation, not a direction?

A 'vector' in $ℝ^2$ has a magnitude and a direction defined on [0, 2$\pi$). How would one describe a similar quantity, but with only an orientation defined on [0, $\pi$)? One example of this sort of ...
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136 views

How to parametrically represent an ARBITRARY circle in polar coordinates?

So, we know that any circle centered at the origin $(0,0)$ can be written in polar coordinates as: $r=f(\theta)=C \iff \langle\theta(t), r(t)\rangle=\langle t,C\rangle$ Where $C\in \mathbb{R}$ is ...
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81 views

What is the spherical equivalent of splitting a circle into n equal segments and calculating their central angles?

So this is easy to calculate in 2 dimensions, if you have a circle represented by 3 points the angle between any two consecutive points and the spheres centre is simply $\frac{2\pi}{n}$. I basically ...
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74 views

The Cramér–Wold theorem for complex random vectors

The Cramér–Wold theorem states that $k$-dimensional real random vectors $X_n$ converge in distribution to a $k$-dimensonal real random vector $X$ if and only if $a^TX_n\xrightarrow{d}a^T X$ for all ...
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127 views

Name for the $\otimes$ operator

I'm teaching 3D vector stuff to engineers. When we write $\mathbf{u} \cdot \mathbf{v}$, we say "u dot v". When we write $\mathbf{u} \times \mathbf{v}$, we say "u cross v". When we write $\mathbf{u} ...
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58 views

If a spider starts from point $A$ and reaches point $B$, find the distance between points $A$ and $B$.

A line $L$ on the plane $2x + y - 3z + 5 = 0$ is at a distance $3$ unit from the point $P(1, 2, 3)$. A spider starts from point $A$ and after moving $4$ units along the line $\frac{x-1}{2}=\frac{y-2}{...
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234 views

Is there a way to prove vector triple product from quaternion multiplication?

For pure imaginary quaternions $u, v, w$, is there a way to prove the vector triple product $u\times(v\times w) = v(u\cdot w) - w(u\cdot v)$ from the relation: $$uv = -u\cdot v + u\times v \text{ for $...
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147 views

Properties and geometrical interpretations of a specific planar vector law.

Genesis of the following thoughts. One usually define the following internal composition law on $\mathbb{R}^2$: $$+:\left\{\begin{array}{ccc} \mathbb{R}^2\times\mathbb{R}^2&\rightarrow&\...
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89 views

Proving a vector identity using Einstein summation

I need to prove $$\nabla \times [(u\cdot \nabla)u]=(u\cdot \nabla)(\nabla \times u)-[(\nabla \times u)\cdot \nabla]u+(\nabla\cdot u)(\nabla \times u)$$ So the ith component of each side using ...
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974 views

Rolling a ball into a cone; what should the forces overall be?

Suppose there is a cone, with the apex pointing down, and the top of the cone at height $h$, apex half-angle $\psi$, ball of mass $m$, and the initial velocity into the cone (completely horizontal, ...
3
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0answers
74 views

Null sum of vectors over the field of two elements

I'm dealing with the test of the International Mathematics Competition for University Students, 2011, and I've had a lot of difficulties, so I hope someone could help me to discuss the questions. I've ...
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66 views

The derivative of a tangent vector having identified vectors as directional derivatives

In introductory calculus, we learn to take derivatives of vectors. For instance, in Cartesian, we have the basis vectors $\vec{e}_x, \vec{e}_y$ and in polars we have $\vec{e}_\rho, \vec{e}_\theta$ ...
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44 views

Is there a name for “almost-subspaces” $U$ that do not satisfy $cu \in U, \forall u \in U$? If so do they have any properties?

I am working with a 'near-subspace' in which the subspace is defined by $$ U := \{p(x) \in \mathbb{P}^3 \mid p(x)> 0, 0\le x \le 1 \lor p(x) \equiv 0 \} $$ This satisfies that $0\in U$ and it ...
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44 views

Origin of the word vector

Lately, I've become interested in the history of vector calculus. The subject is generally considered to have come into existence with the work of Hamilton and Grassmann in the 1840s to 1860s. The ...
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110 views

Prove that the cross product of two vectors is a vector

I define a vector as any object $(a_i,a_j, a_k)$ such that it transforms the same way as the coordinates themselves. That is if $x'_i = R_{ij}x_j$, then $a'_i = R_{ij}a_j$ (using the Einstein ...
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48 views

Apostol Calculus I-14.19 exercise 16(missiles problem)

Problem:Due to a mechanical failure,a ground crew has lost control of a missile recently fired.It is known tha the missile will proceed at a constant speed on a straight course of unknown direction....
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108 views

How to find optimal solar panel angle using vectors?

Basically I am trying to find the optimum angle at which a solar panel should be installed by using vectors. I have done some research but found it a bit confusing , so basically I haven't got very ...
3
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0answers
53 views

Name for a vector that cannot be removed from a set without changing the span

Consider the set $$S=\left\{\begin{bmatrix}2\\3\\4\end{bmatrix}, \begin{bmatrix}1\\2\\0\end{bmatrix}, \begin{bmatrix}2\\4\\0\end{bmatrix}\right\}$$ Either the 2nd or 3rd vector can be removed from $...
3
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0answers
52 views

Please explain how the following derivative graphically makes sense.

I have two vectors $\vec{A}$ and $\vec{B}$ as shown below: The point at the origin of vector $\vec{B}$ has coordinates $(x,y)$. The angle between the two vectors is $\theta$. Now in my physics book ...
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0answers
95 views

Is There a Unitary Transformation Which Maps Non-Orthogonal Vectors into 'Uniform' Vectors?

I would like to find a unitary matrix $U$ which maps normalised vectors $\mathbf{v}^{(i)}$ from the set: \begin{equation} \mathcal{V} = \Bigg\{ \begin{pmatrix} 1 \\ 0\\ 0 \\ \vdots \end{pmatrix}, \...
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48 views

Probability with replacements

Three components of vectors are chosen randomly from the digits $0$ to $5$ with replacements. Prove/disprove that $$P(\text{magnitude }=5)=\frac 1{24}$$ Here's my working. Please correct me if ...
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65 views

Geometric Interpretations of the Cramer's Rule

In Cramer's rule when $Δ_x=Δ_y=Δ_z=Δ=0$ what are the possibilities for the three planes (three simultaneous equations in $x,y,z$) to look like? My book says that they can have either infinite ...
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0answers
117 views

are all vectors position vectors?

So i have two questions: 1) If Vectors (say in R2) are independent of their location in space, then couldn't you theoretically for any vector in R2 regardless of its position, just move it such that ...
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46 views

Zero Vectors and Vector Spaces

In a solution guide I've read that $\begin{bmatrix}a\\b\\c \end{bmatrix} : a+b+c=2$ isn't a vector space because it doesn't have the zero vector. I'm not 100% sure why this is but I think it's because ...
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50 views

Number of Bases in $C^8$

Consider the set of all the vectors in $C^8$ each of whose coordinates is either 0 or 1. How many different bases does this set contain?
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160 views

Find all scalars $\lambda$ for which the vectors $\vec{a}, \vec{b}, \vec{c}$ are coplanar.

Given the vectors $\vec{a}=\vec{i}+2\vec{j}-\vec{k}, \vec{b}=\vec{i}-\vec{j}+\vec{k}, \vec{c}=\lambda\vec{i}+\vec{k},\lambda\in\Bbb R$ in $V^{3}(0)$ find all scalars $\lambda$ for which the vectors $\...
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99 views

How to prove that one formula is numerically better than another

If $\mathbf{u}$ and $\mathbf{v}$ are vectors in real 3-dimensional space, here are two formulas for computing the angle between them: $$\theta = \operatorname{atan2}\left( \|\mathbf{u}\times\mathbf{v}...
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102 views

Are two dot products of a random variable vector independent?

Let $w,v$ be two different vectors in the finite vector space $Z_p^m$ over $Z_p$ where $p$ is prime. Let $u$ be a vector chosen uniformly at random from $Z_p^m$. Are the random variables $u \cdot w$ ...
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127 views

Find the ratio of the area of $\triangle ABC$ to area of $\triangle AOC$

Let $O$ be an interior point of $\triangle ABC$ such that $2\vec{OA}+5\vec{OB}+10\vec{OC}=\vec{0}$.Find the ratio of the area of $\triangle ABC$ to area of $\triangle AOC$. My Attempt: Let $O$ be the ...
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0answers
30 views

Maximum number of vectors with $\mathbf{a_i} \cdot \mathbf{a_j} < 0$

Find the maximal $k$ such that there exist vectors $\mathbf{a_1},...,\mathbf{a_k}$ with the property that $\mathbf{a_i} \cdot \mathbf{a_j} < 0$ for all $1 \leq i < j \leq k$ if all vectors are ...
3
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0answers
59 views

How can i reflect position and direction vectors from a plane

I'm now working on a project that has mirrors. I'd like to reflect a virtual camera and the way which i can do this is to reflect two vectors - position and normalized direction vectors of the camera. ...
3
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0answers
95 views

Vectorization of the nonzero entries of a matrix

I am familiar with the $\textrm{vec}(A)$ operator, where the columns of a matrix $A$ are stacked into a vector. If my matrix $A$ has zero-entries, is there a standard notation/operator like $\textrm{...
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0answers
150 views

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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0answers
198 views

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 ...
3
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0answers
34 views

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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0answers
55 views

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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0answers
71 views

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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0answers
206 views

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 ...
2
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0answers
30 views

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 ...
2
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0answers
44 views

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 ...
2
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0answers
93 views

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 ...
2
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0answers
32 views

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 ...
2
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0answers
42 views

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 ...