# 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
Filter by
Sorted by
Tagged with
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 ...
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 ...
72 views

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 $[f(x_1),\... 0answers 228 views ### are all vectors position vectors? So i have two questions: 1) If Vectors (say in R2) are independent of their location in space, then couldn't you theoretically for any vector in R2 regardless of its position, just move it such that ... 2answers 894 views ### Find the limit of the vector as$~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 + \...
477 views

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.
200 views

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 ...
29 views

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 ...
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 ...
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 ...
96 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 ...
169 views

280 views

100 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 ...
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 ...
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 ...
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, ...
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 ...
43 views

44 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 ...
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: ...
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 ...
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 ...
82 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 ...
96 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 $\vec{e}_\rho, \vec{e}_\theta$ ...
127 views

45 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 ...
39 views

84 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 ...
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 ...
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 ...
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}, \...
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. ...
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 ...
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 \...