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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Minimizing the sum of cosines of non-obtuse angles formed by $n\geq4$ concurrent lines in $3$D space

Suppose I have two lines in $3$D space passing through the origin. The smallest angle formed between them would be between $0$ and $\pi/2$. Minimizing the cosine of this angle we'll get $\cos {(\pi/2)}...
Dotman's user avatar
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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 ?
Laurent Crivello's user avatar
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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 ...
MathLearner's user avatar
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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 ...
Stan Shunpike's user avatar
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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 ...
Aaryan Garg's user avatar
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A few questions on Linear Algebra

I actually posted this question a few weeks ago where I wanted my solutions to a few Linear Algebra questions checked. Now thanks to useful links provided by @GerryMyerson I can ask my questions (and ...
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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....
samer abdullah's user avatar
5 votes
1 answer
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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),\...
saro falsaperla's user avatar
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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} ...
bubba's user avatar
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A practical example of Helmholtz decomposition

I am familiar with the basic concept of the Helmholtz decomposition and I have read a number of materials on it (they all follow structure similar to that on Wikipedia page). However, I am not able ...
nevermind's user avatar
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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|\}...
user1963's user avatar
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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 ...
user2272296's user avatar
5 votes
3 answers
120 views

Finding the angle between vectors

What is the difference between $\cos(\theta) = \frac{(v \cdot u) }{( \|v\| \|u\| )}$ and $\cos(\theta) = \frac{|(v\cdot u)|}{( \|v\| \|u\| )}$ I noticed that my textbook uses simply the value of the ...
Jed's user avatar
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How to prove that the functions cot(x), cot(2x), ..., cot(nx) are linearly independent?

I'm trying to prove that the functions: $cot(x), cot(2x), \dots, cot(nx)$ are linearly independent. My idea was to use mathematical induction, that is: For $n = 1, \hspace{0.5cm} \alpha_1 cot(x) \...
John's user avatar
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A system of three `eigenequations' (sort of)

In my research I have stumbled upon the following problem: Find all pairwise orthonormal triples of vectors $\mathbf{x},\mathbf{y},\mathbf{z} \in \mathbb{C}^n$ ($n \geq 4$) satisfying the three `...
meler's user avatar
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Reference request: A primer on the difference between "Vectors" in linear algebra vs. "Vectors" in ohysics/vector calculus

(This question is inspired by a recent question on this site.) The word "vector" means different things in different contexts. Most students who encounter the term first learn it in the ...
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A navigation problem: Is the path of the ship straight or curved?

My first post here: I’m looking for some guidance with a maths problem. A ship sets sail from England (A) to France (B) covering a distance of 20 miles at an average speed of 5mph. If the ship sails ...
Stephen Clement's user avatar
4 votes
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93 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 ...
Cat's user avatar
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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 ...
Florian Ente's user avatar
4 votes
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362 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, ...
Robert Rendell's user avatar
4 votes
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1k 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{...
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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 ...
LudvigH's user avatar
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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{...
bertozzijr's user avatar
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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 ...
Kim Fierens's user avatar
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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 ...
S E Clark's user avatar
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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 ...
Makogan's user avatar
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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 ...
Dimitrios Efthymiou's user avatar
4 votes
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162 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 ...
Cm7F7Bb's user avatar
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How to rotate / transform a vector so that it is parallel to a plane?

My background: I have taken an introductory class or two in linear algebra at university, but it was a few years ago and I have not practiced it since. My terminology might be off, but I believe my ...
user1323245's user avatar
4 votes
0 answers
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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}{...
MathGeek's user avatar
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4 votes
1 answer
152 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}...
JWWalker's user avatar
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4 votes
0 answers
156 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&\...
C. Falcon's user avatar
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4 votes
1 answer
909 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 ...
user225683's user avatar
4 votes
0 answers
874 views

Standard Deviation of 2D vector data

Given a sample set of wind data (speed and direction parallel to the earth), I would like to identify the consistency of wind samples. Standard deviation comes to mind, but I don't know if it is ...
Cowdozer's user avatar
4 votes
0 answers
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, ...
Sam Walls's user avatar
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What is the opposite of antiparallel?

Is parallel really the opposite of anti-parallel? While anti-parallel indicates that two vectors are directed along the same line and are pointing in opposite directions then it seems to me that ...
Steeven's user avatar
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172 views

Find the indeterminate values of $x_1$ and $y_1$ if $\vec x=(x_1,-2,1,-1)$ and $\vec y=(-2,y_2,-1,-2)$ and $\lVert \vec x\rVert=2\lVert \vec y\lVert$

Let $\vec x=(x_1,-2,1,-1)$ and $\vec y=(-2,y_2,-1,-2)$, wich satisfies $\lVert \vec x\rVert=2\lVert \vec y\lVert$. Find the indeterminate values of $x_1$ and $y_1$. So, assuming $\lVert _\dot{} \lVert ...
Roma_Rayado's user avatar
3 votes
1 answer
292 views

Why cross product gives area of parallelogram formed by two vectors

Recently we were introduced to the concept of vectors in our class, and we learnt about dot products and cross products.I do know that $a\times b$ yields the area of the parallelogram formed by the ...
Aniket Harit's user avatar
3 votes
0 answers
48 views

A function on binary vectors

I am looking for a function $f:\{0,1\}^d \to \{0,1\}^{d'}$, where $d'<5d$, such that whenever $x,y,z$ are three distinct $d$-dimensional binary vectors, and for all $i\in\{1,\dots, d\}$ it holds ...
Freshman's Dream's user avatar
3 votes
0 answers
41 views

How do I apply the distributive law for the vector equation?

How do I apply the distributive law for the equation $$xA - xBA = 0$$ where $x$ is a vector, $A$ and $B$ are matrices? I can see two ways of doing this but I can't see how both of them can be true at ...
Atul Ramkrishnan's user avatar
3 votes
0 answers
48 views

Rotating a 3D shape so that it gets heated evenly by a fire

Imagine you have a shape (say, an eggplant) that you want to cook roughly evenly on a fire. How should you rotate the eggplant to accomplish this? More concretely, the surface of the eggplant (before ...
chausies's user avatar
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3 votes
0 answers
145 views

Rodrigues equation VS Hamilton's quaternions. Historical confusion.

I don't have a special math education and when I study quaternions I spent a long time. Along the way, without realizing, I independently derived the Rodrigues equation, because now we know about the ...
OpenglNoob's user avatar
3 votes
0 answers
125 views

Solving an equation involving a matrix exponential

Suppose we have unknown scalars $x_1, x_2, ...,x_m \in \mathbb{R}$, known matrices $A_1, A_2, ...,A_m \in \mathbb{R}^{n\times n}$, and two known vectors $s_0, s_1\in\mathbb{R}^n$. I want to find $x_1,...
Ran's user avatar
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3 votes
0 answers
91 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 ...
wamitw's user avatar
  • 56
3 votes
0 answers
230 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 ...
Givi Odikadze's user avatar
3 votes
0 answers
103 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{...
Ali Alidoust's user avatar
3 votes
0 answers
224 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 ...
Max Muller's user avatar
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3 votes
0 answers
147 views

Is there a version of tuples without duplicates?

A set is a list of elements that is unordered and does not permit duplicates, so: $$\{1,2\} = \{2,1\}$$ $$\{1,2\} = \{1,2,2\}$$ A bag or multiset is unordered, but it allows duplicates, so: $$[1,2]=[2,...
Nirvana's user avatar
  • 447
3 votes
1 answer
786 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{...
Anon21's user avatar
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3 votes
0 answers
70 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 ...
Anon21's user avatar
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