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.

2,839 questions with no upvoted or accepted answers
Filter by
Sorted by
Tagged with
7
votes
2answers
729 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.
7
votes
0answers
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 ...
6
votes
0answers
67 views

How to arrange vectors in a circle so as to minimize the resultant?

Sorry for the vague language, but I want to arrange weights in a circle so as to minimize the resultant moment. The axis of rotation is perpendicular to the plane in which the weights lie, and the ...
6
votes
0answers
284 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 ...
6
votes
0answers
1k 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 ...
5
votes
0answers
86 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....
5
votes
1answer
74 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),\...
5
votes
0answers
309 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 $...
5
votes
2answers
1k 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 + \...
5
votes
0answers
201 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|\}...
5
votes
0answers
7k 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 ...
4
votes
0answers
52 views

Interpreting Angles in Higher Dimensions

I am measuring the angles between vectors in the $n$-dimensional non-negative orthant (i.e., the non-negative subset of $\mathbb{R}^n$). $n$ is large. I am wondering how much I can interpret angles in ...
4
votes
0answers
92 views

Matching a target velocity and position

For a science-fiction computer game I'm writing, I wish to calculate the travel time and accelerations of a ship leaving one (3D) position with an initial velocity, and travelling to a moving target, ...
4
votes
1answer
209 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^{...
4
votes
0answers
825 views

Angles between vectors using a non-standard inner product

What is the angle between $\vec{x}=\begin{bmatrix} 1 \\ 1 \end{bmatrix}$ and $\vec{y}=\begin{bmatrix} -1 \\ 1 \end{bmatrix}$ using the inner product defined as $⟨\vec{x},\vec{y}⟩=\vec{x}^{T}\begin{...
4
votes
0answers
31 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{...
4
votes
0answers
83 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 ...
4
votes
0answers
365 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
votes
1answer
530 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 ...
4
votes
0answers
104 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 ...
4
votes
0answers
232 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} ...
4
votes
0answers
67 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}{...
4
votes
0answers
120 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}...
4
votes
0answers
151 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&\...
4
votes
0answers
113 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 ...
4
votes
1answer
166 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 ...
4
votes
1answer
817 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 ...
4
votes
0answers
1k 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
votes
0answers
24 views

Contractive Fixed Point Theorm for Vector Value Functions

Consider the following question: I tried to look on the function $g(x)=f(x)+[x_1,x_2]$ , so if I could prove that $g(x)\in S$ for $x\in S$, then by the contractive fixed point theorm there exists ...
3
votes
0answers
54 views

Comparing angles of high dimensional vectors

I'm using cosine similarity to compare angles of 10000 dimensional vectors. This works fine, but I'm wondering if it's possible to instead, store the angles between certain vectors and a unit vector ...
3
votes
1answer
38 views

Inequality for $p$ vector norm

Is it true for $2 \leq q \leq p$ that $$ \|x\|_q\leq n^{\frac{p-q}{pq}} \|x\|_p $$ where $x$ is an $n$-dimensional vector. I only need the inequality for $n=2$, so that would suffice. I'm just curious ...
3
votes
0answers
73 views

Angle between two planes in $n$-dimensional space, where $n \geq 4$

As known, an angle between two planes in $3$d is an angle between normals to them. But how about $n$-dimensional spaces, $n\geq4$. Is it the same for them? Why is it so? I have two planes, given ...
3
votes
0answers
84 views

N-Body Simulation in a Seamlessly Repeating World

For running an N-Body simulation it is required to calculate the force between every pair of massive bodies. The force applied on body $a$ from body $b$ is calculated as follows: $$F_{ab} = -G\frac{...
3
votes
0answers
180 views

Does the functional square root of the cosine admit a vector-based interpretation?

In linear algebra, the cosine of the angle between two vectors $a$ and $b$ is defined as $$\cos(a,b) = \frac{\langle a, b \rangle}{||a||\cdot||b||} .$$ The functional square root of the cosine has at ...
3
votes
0answers
281 views

What does the exponential of a vector do geometrically?

The exponential of an even multi-vector is related to rotation, but what is the exponential of a vector? For instance, the exponential of a vector $\mathbf{v}=x\hat{\mathbf{x}}+y\hat{\mathbf{y}}+z\hat{...
3
votes
1answer
228 views

Geometric Intuition of the Dot Product

First of all, sorry for my poor English and thanks for your time. I’m having problems to understand the intuition behind the dot product. I know how to calculate the dot product with the algebraically ...
3
votes
0answers
55 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 ...
3
votes
0answers
47 views

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,...
3
votes
1answer
42 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}=(...
3
votes
0answers
45 views

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 ...
3
votes
0answers
26 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: ...
3
votes
1answer
266 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 ...
3
votes
0answers
63 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
votes
1answer
219 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 ...
3
votes
0answers
1k views

In what spaces are outer product of x with itself positive semidefinite?

In the case of vectors in $\mathbb R^n$, it is quite simple to see that for any vector $x$, $$ v^Txx^Tv = (v^Tx)^2 \geq 0$$ so clearly the form $xx^T$ must form a positive semidefinite matrix. But ...
3
votes
0answers
85 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 ...
3
votes
0answers
103 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 $\vec{e}_\rho, \vec{e}_\theta$ ...
3
votes
0answers
189 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,...
3
votes
0answers
46 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 ...
3
votes
0answers
351 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 ...

1
2 3 4 5
57