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.

2,541 questions with no upvoted or accepted answers
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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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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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1answer
72 views

Full rank condition when stacking vector valued function

Let $f: \mathbb{R}^1 \mapsto \mathbb{R}^n$ be a smooth vector-valued function. Consider the $n \times n$ matrix $A(x)$ obtained from a vector $x \in \mathbb{R}^n$ by appropriately stacking $[f(x_1),\...
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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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2answers
894 views

Find the limit of the vector as $~t~$ approaches $~0~$?

Here's the vector: $$e^{-6t}~\vec i + \frac{t^2}{\sin^2t}~\vec j + \sin(6t)~\vec k$$ Don't I just take the limit as $t$ approaches $0$ from each individual component? As a result, I got $~\vec i + \...
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477 views

About Cauchy–Schwarz inequality

For the vectors $x$ and $y$, the Cauchy–Schwarz inequality reads $$ |x\cdot y|\leq||x||\cdot||y|| $$ Does this inequality only hold for 2-norm? Or for any norms? Thanks in advance.
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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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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 ...
4
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1answer
93 views

How can I find the modulus of the angular acceleration when an instantaneous acceleration is given?

The problem is as follows: A pulley starts spinning from rest a rotation with constant angular acceleration. After $5\,s$ a point in its periphery has an instant acceleration which makes a $53^{...
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136 views

Norm of the sum of random vectors from a unit ball

Let $x_1,\dots, x_n\in \mathbb{R}^d$ be independently distributed from a uniform distribution on a ball of radius $1$. That is for every $i$: $x_i \sim U(\{x\in \mathbb{R}^d: \|x\|_2\leq 1\})$. We ...
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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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183 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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82 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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246 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 ...
4
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1answer
430 views

How to interpret units of measurement for vector magnitude?

If we take a Cartesian system with length over length then the vector magnitude has length as unit of measurement. I get that. Let's say we have cups of coffee as the x axis and cost in dollars as the ...
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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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169 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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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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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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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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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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1answer
126 views

Analogs to vectors — *unoriented* line segments

A real vector can be thought of as an oriented line segment. Linear algebra and multivariable calculus can be taken pretty far just by considering these types of objects (obviously there are ...
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1answer
742 views

Equation of a plane containing a straight line

A straight line passes through the points $(1, 2, 3)$ and $(-3, -2, -1)$. I have calculated the system of equations of this line to be $$ x = 1 - t,\, y = 2 - t, \, z = 3 - t $$ The question I ...
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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, ...
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46 views

How can I bend a complex number (and multivectors in general)?

Let me use the following notation for orthonormal basis $\{\sigma_0,\sigma_1,\dots\}$ and this one for a general curvilinear basis $\{\mathbf{e}_0,\mathbf{e}_1,\dots\}$. The basis are constrained by ...
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A combinatorial problem of choosing from vectors of length at least 1

$v_1,v_2,\dots,v_n\in \mathbf R^2$ are $n$ 2-dim vectors of length at least 1. Show that we can choose at most $m={n\choose \lfloor n/2\rfloor}$ subsets of $\{v_1,v_2,\dots,v_n\}$ , say $S_1,S_2,\dots,...
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1answer
37 views

Find $\vec{r}$ satisfying the equation $\frac{d^2\vec{r}}{dt^2}+\omega^2\vec{r}=0$

Problem : Find $\vec{r}$ satisfying the equation $\dfrac{d^2\vec{r}}{dt^2}+\omega^2\vec{r}=0$, where $\omega$ is a constant different from zero. Given that $\vec{r}=(1,1,1)$ and $\dfrac{d\vec{r}}{dt}=(...
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How to compute similarity between users given their ratings?

I'm going to use a hypothetical Netflix as an example. Let's pretend they have 1,000,000 users and allow users to rate shows from 1 to 5 stars, and there are 500 shows total. Let's also say the users ...
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24 views

Switch 2 elements in a 2d vector algebraically

I'm new to lineair algebra so maybe I am asking a silly question here. What I mean with the title is this command in Matlab/Octave: ...
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43 views

Is it possible to calculate the shift vector of two vector?

I'm working on some image processing tasks and I'm required to calculate the 'shift vector' of two feature vectors. I'm not entirely sure what 'shift vector' really means as i just came across it, but ...
3
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1answer
109 views

Maximum number of “almost orthogonal” vectors in a complex vector space

In a $d$ dimensional vector space defined over $\mathbb{C}$, how do I calculate the largest number $N(\epsilon, d)$ of vectors $\{V_i\}$ which satisfies the following properties. Here $\epsilon$ is ...
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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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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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2answers
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Geometric visualization for volume of parallelepiped

Volume of parallelepiped defined by vectors \begin{align} \begin{bmatrix}a\\b\\c\end{bmatrix}&,\quad \begin{bmatrix}v_1\\v_2\\v_3\end{bmatrix},\quad\begin{bmatrix}w_1\\w_2\\w_3\end{bmatrix} = \...
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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 ...
3
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1answer
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Part of proving a set is a subspace of $\mathbb{R}^3$

Say I have the set $S=\{\bar{x}\in\mathbb{R}^3:x_1+x_2=0\}$ One of the things I have to prove is that every two vectors in $S$, their sum is also in $S$ Does the following prove that?: Let $\bar{a},...
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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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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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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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help with a vector proof

Let $\textbf{u}$ and $\textbf{v}$ be non-zero vectors. Show that any two-dimensional vector can be expressed in the form $$s \textbf{u} + t \textbf{v},$$where $s$ and $t$ are real numbers, if and only ...
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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 ...
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75 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 $...
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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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53 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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272 views

Outer product of cross product vector with itself

I'm wondering if there is another, possibly more efficient, way to get to the $3 \times 3 $ symmetric matrix $\mathbf{D}$ below from 3-vectors $\mathbf{a}$ and $\mathbf{b}$ then the straight forward ...
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130 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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49 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 wrong. ...
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83 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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Angles of a known 3d vector

I have a 3d vector r known by its coordinates rx, ry, rz. How can calculate angles Theta and Phi ?
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1answer
41 views

What does this vector, which is a combination of points in triangle, mean geometrically?

Suppose $\vec{a}=\overrightarrow{\textrm{BC}}, \vec{b}=\overrightarrow{\textrm{CA}}, \vec{c}=\overrightarrow{\textrm{AB}}$ for a triangle $\triangle \rm{ABC}$ in a plane. Let $\vec{p}=(\vec{a} \cdot \...

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