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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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Where does this formular for rotating a vector in 3D space around another 3D vector comes from?

I found this formular: $\mathbf{R}_{\vec{n}}(\alpha)\vec{x}=\vec{n}(\vec{n}\cdot\vec{x})+\cos(\alpha)(\vec{n}\times\vec{x})\times\vec{n}+\sin(\alpha)(\vec{n}\times\vec{x})$ here: https://de.wikipedia....
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0answers
7 views

Properties of outer product of two unit vectors? Why is there only one non-zero eigenvalue for such a matrix?

Let $x,y$ be two unit vectors. $A=xy^T$ be the outer product. The eigenvalues of A are seen to be $[0, 0, 0,...0, k]$. Why is that? What are the properties of the outer product of two unit vectors? ...
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1answer
32 views

What does it mean for a function from R^n to R to be convex?

I'm looking to determine when the function $f(\vec{x}) = k\vec{x}\cdot\vec{x} - \vec{x}\cdot\vec{y}$ is convex. However, I'm not even sure where to start. For a function $\mathbb{R} \rightarrow \...
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0answers
16 views

Computing the shape operator $S_\eta :T_pN \rightarrow T_pN$

Let $f: \mathbb{R}^2 \rightarrow \mathbb{R}^3$ be defined by $$f(x, y) = (\text{cos}(x)(\text{cos}(y)+4), \text{sin}(x)(\text{cos}(y) +4), \text{sin}(y)).$$ How would I go about computing the shape ...
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2answers
27 views

Component-Wise 3D Vector Projection

I want to define the projection of a vector $\mathbf{v} \in \mathbb{R}^3$ onto the line $\mathbf{r} \in \mathbb{R}^3$ in terms of the components of $\mathbf{v} = (v_x, v_y, v_z)$. In 2D, this looks ...
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1answer
29 views

Transform coordinate system for the gradient of a function at a specific x, y, z value.

I have the gradient of a function $f(x,y,z)$ at a specific value of $x, y,$ and $z$ in the vector form: $$ \nabla f(x,y,z)\Bigr|_{\substack{x=x_1\\y=y_1\\z=z_1}} =\begin{pmatrix} \frac{\partial{f}}{\...
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0answers
15 views

Quantifying positining and alignement of two line segments

For simplicity we can consider a 2D case: where we have line segments of the same length $l,$ and we suppose we have $n$ of them with their centre coordinates and orientations randomly assigned. For ...
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0answers
14 views

Finding vector potential from a given magnetic field

I want to find the vector potential from a given magnetic field in three-dimension in cartesian coordinates for two cases. 1) Where the magnetic field is in any analytical form and 2) When the ...
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0answers
37 views

The commutator product as a replacement for the cross and wedge product in geometric algebra?

From an axiomatic approach to geometric algebra, the wedge product of two vectors $a$ and $b$ is typcially defined as the antisymmetric $a \wedge b = \frac{1}{2}(ab - ba)$, where $ab$ is the geometric ...
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0answers
22 views

Vector multiplication and normalisation

I am studying for a paper, and would like to understand if the following operation represent a normalisation. I have two vectors A, B of same length. I multiply their items, element-wise, sum the ...
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1answer
24 views

how to represent a vector [on hold]

I have to define a state space equation. there is a vector with 5 members as stats such that all states(members of the vector) are real numbers. which notation is true for this case? x=[x1,x2,x3,x4,...
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1answer
34 views

Matrix and vectors (Find 3 Vectors)

I am having a bad time with this matrix and vector situation, and I think the solution is kind a trick in some part of the computation, but I don't know how to find this: Find 3 vectors (different ...
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1answer
37 views

How to prove $v_1\cdot v_2=|v_1||v_2|\cos(\theta)$ in n-dimensions?

It is easy to prove in 2D that $v_1\cdot v_2=|v_1||v_2|\cos(\theta)$ where $\theta$ is the angle between $v_1$ and $v_2$. But how to generalize? What is the proof in n-dimensions?
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2answers
27 views

Why is the angle between vectors restricted?

Why is the calculated angle between two vectors always between $\pi$ and $0$. Is this due to the limitations of $\arccos\theta$ or is it because angles between vectors is described to be the smaller ...
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1answer
39 views

Find a and b and the position vector of the point of intersection of C of $l_1$ and $l_2$

I asked a similar question to this yesterday, and I think I managed, however, I have a similar question but a bit different, if I understand this I think I'll manage to confirm the other one as well, ...
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1answer
33 views

Using partial derivatives to find normal vector

So, I can not find out what I'm doing wrong with this question, even if my life depended on it. I know instruction for doing it, but I can't seem to figure out what I'm doing wrong. Because I refuse ...
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0answers
16 views

Questions about the basics of coordinate systems and their basis

Since I think I am missing some basic understanding about coordinate systems and their basis, I would really appreciate your help answering my questions. Also if something isn't written mathematically ...
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1answer
42 views

Trying to prove an integration by parts formula

Denote pde operator $$ Lu = - div \cdot (p \nabla u ) + qu$$ where $x \in D$ and $p=p(x) > 0$ and q=q(x) are continuous on $\bar{D}$ an p has continous first partial derivatives on $\bar{D}$. I ...
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2answers
54 views

Find a and b given that $l_1$ and $l_2$ intersect at the midpoint of the line segment $\vec{AB}$

A past examination paper had the following question that I found somewhat difficult. I tried having a go at it but haven't come around with any solutions. How would one go about tackling it? The ...
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1answer
22 views

Coordinates of 3D vector in rotated coordinate system (without using a matrix)

The problem: There is a vector with coordinates X,Y,Z. This vector is in a coordinate sytem that has been rotated by A degrees along the X axis and B degrees along the Y axis. I would like to know ...
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1answer
36 views

Clarification in proof of perpendicular bisectors meeting at a point

Context: My crude drawing from Paint to illustrate triangle OAB: My working: $\begin{align*} (z-\frac{1}{2}(x+y)) \cdot (y-x) &= z\cdot(y-x) + \frac{1}{2}(-x-y) \cdot (y-x)\\ &= z\cdot y -z\...
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0answers
11 views

Multiplication of Phasors as Complex Numbers and Vectors

I am reading about phasors. Everywhere it says that phasors are complex numbers and vectors which is obvious given that every complex number is a two-dimensional vector. But there are a lot of ...
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1answer
43 views

Simplifying $|\vec{A} \times (\vec{B} \times \vec{C})|^2$

I know the vector identity $\vec{A} \times (\vec{B} \times \vec{C}) = (\vec{A} \cdot \vec{C}) \vec{B} - (\vec{A} \cdot \vec{B}) \vec{C}$ Now, is there a succinct way of obtaining $|\vec{A} \times (\...
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0answers
12 views

Simplify $c_i=a_j\frac{\partial b_j}{\partial x_i}$

Consider two 3-D vector fields $\vec{a}$ and $\vec b$. Then define a new vector field $\vec c$ as $$ c_i = a_j\frac{\partial b_j}{\partial x_i}, $$ i.e., $$ \vec c = \left( \vec a \cdot \frac{\...
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5answers
54 views

Prove that $u\cdot v = \frac{1}{4}||u+v||^2 - \frac{1}{4}||u-v||^2 \forall u,v \in \mathbb{R^n}$

I am trying to prove the above statement but I'm not sure if my proof is correct. My proof is as follows, Given $u\cdot v$, we know by the C-E Inequality that $|u \cdot v| \leq ||u|| \ ||v||$ ...
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0answers
16 views

Erroneous vector filtering from image based on direction (directional filter) [on hold]

I am working on the motion estimates using two images by python based algorithm. I got the attached vector field (green color). This field includes some erroneous vector. I want to filter those ...
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1answer
15 views

How to calculate the vector intersecting a sphere tangent and plane

I have a sphere centred at a point (x, y, z) = (0, 0, 0) with radius r = 1. I have a point P on the outside of the sphere. How could I calculate a vector at P, which points along both the tangent ...
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1answer
22 views

Given: $u, v \in \mathbb{R}^3$ find all vectors $w$ such that $u \cdot v = w \cdot u$ [closed]

I have the following problem, but I do not know how to approach it. Given: $u, v \in \mathbb{R}^3$ find all vectors $w$ such that $u \cdot v = w \cdot u$ (dot product) Can anyone give me a hint on ...
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3answers
51 views

Proof that $\| u \|=\| v\|\iff\langle u+v,u-v\rangle=0$

Let $(\cdot,\cdot):\mathbb{R}^n\times \mathbb{R}^n\to\mathbb{R}^n$ be any map that fulfills the properties $u,v,w\in\mathbb{R}^n;\; \lambda\in\mathbb{R};\; \langle u;v\rangle:=(u_1v_1)+\cdots+(...
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1answer
36 views

Vectors dot product

Could someone please spot the error I've made in this question: Q: A theme park has two zip wires. Sarah models the two zip wires as straight lines $(L_1, L_2)$ using coordinates in metres. The ends ...
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1answer
36 views

How can we justify $\vec F = \frac{d \vec p}{dt}$ by just vector subtraction? [closed]

I want to justify Newton's second law for the linear momentum of a particle: $$\vec F = \frac{d \vec p}{dt}$$ using really basic linear algebra. Basically, just with vector subtraction. This is ...
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0answers
37 views

Subtraction of vectors in spherical space

I have 4 microphones placed in a spherical coordinate system. I know the $(r_i,\theta_i,\varphi_i)$ for each microphone $m_i$. Given a speed of sound $C$ and the direction from which the sound arrives ...
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0answers
24 views

Write a sequence of vectors as a matrix

Let $X \in \mathbb{R}^{N\times T}$, where $N,T\in \mathbb{Z}^{> 0}$, and $x\in (\mathbb{R}^N)^T$, by which I mean $x = (x_1, x_2, \dots, x_T)$, where $x_i \in \mathbb{R}^N$. The objects $X$ and $x$...
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1answer
28 views

Linear span of given vectors expresses a plane in $\mathbb R^3$?

Given vectors $(1, 3, 5), (-2 , -6, -10)$ and $(2, 6 , 10)$ determine whether the linear span of the above is a plane in $\mathbb R^3$. The vectors are linearly dependent nd hence do not form ...
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0answers
21 views

Signed angle between higher-dimensional oriented vectors?

I am working with vectors in $\mathbb{R}^4$. Any two such (non-parallel) vectors obviously define a plane, and I can rotate any vector in the plane defined by itself and a second vector as follows: $...
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1answer
41 views

How to get a basis of $U \cap V$ where U and V are the column space of $A$ and $B$.?

Problem: Let $A = \begin{bmatrix}5& 2& -1\\3& 1& 0\\ -1& 0& -1 \end{bmatrix}$, $B = \begin{bmatrix} 4& -3\\ -2& 3\\ 1& -2\end{bmatrix}$, $U = C(A)$ and $V = ...
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0answers
15 views

Finding two unknown scalars using vectors GCSE

Asking about question 9d) in attached photo. I have the mark scheme to this question (also attached) and I understand the principle of 9d). I think you have to equate it somehow, using the ...
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1answer
28 views

Linear dependence of three vectors

Could someone please explain why, if two vectors in a set of three are parallel to each other, that this implies that the whole set of three vectors is linearly dependent? I have tried to show this ...
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2answers
60 views

Is a quaternion a way to divide vectors? [closed]

This is a naive question but I read popular science articles where it is stated that quaternions define vector division, without further explanations
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1answer
52 views

Determining value of product $ (2\vec{a}+\vec{b}).[(\vec{a}\times\vec{b})\times(\vec{a}-2\vec{b})]$ [closed]

enter image description hereenter image description hereenter image description hereIf $\vec{a}$ and $\vec{b} $ are vectors in space given by $\vec{a} = \frac{\hat{i}-2\hat{j}}{\sqrt{5}}$ and $\vec{b}...
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1answer
37 views

vectors and scalars… [closed]

If $\vec u=\hat i×(\vec a×\hat i)+\hat j×(\vec a×\hat j)+\hat k×(\vec a×\hat k)$, then: (A) $\vec u$ is a unit vector (B) $\vec u=\vec a+\hat i+\hat j+\hat k$ (C) $\vec u=2\vec a$ (D) $\vec u=...
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0answers
15 views

Understanding the orthogonal projection vector derivation

As you can see below, $z$ is the projection of $x$ onto $y$... I am trying to derive the orthogonal projection formula based on things I already know. Calculating $cos(\theta)$ is trivial... $$cos(\...
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0answers
31 views

How to find the rotation vector by deriving the final vector with respect to the displacement?

My understanding of a rotation of a vector can be done by using a 2D rotation matrix as shown below, $R(\theta )=\begin{bmatrix}\cos \theta &\sin \theta \\-\sin \theta &\cos \theta \\\end{...
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2answers
48 views

Show that a vector is equal to the reciprocal of its reciprocal

Given three vectors $\vec u$, $\vec v$, $\vec w$, and their reciprocals $$ \vec u^{\,\prime} := \frac{\vec v \times \vec w}{(\vec u,\vec v, \vec w)} \qquad \vec v^{\,\prime} := \frac{\vec w \times \...
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0answers
37 views

Which of the following are true for the vector $\overrightarrow A$? [closed]

The vector $\overrightarrow A$ has magnitude $A$ and $\hat A$ is a unit vector in the direction of $\overrightarrow A,$ then which of the following are correct: $1)$ $\overrightarrow A \cdot \hat A= ...
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0answers
6 views

Unit vector Identicon

Is there a way to visualize multiple n-dimensional ($n\approx300$) unit vectors with the property that large changes to the vector result in large visual changes, but small changes result in little to ...
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1answer
39 views

How to find a vector perpendicular to plane ABC given A,B and C

Let $A=(1,0,1)$, $B=(2,1,-1)$, $C(0,1,2)$ Find a vector perpendicular to the plane $ABC$. the solution I was given by my lecturer: Does it matter which vectors I use? Because my attempt has got ...
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1answer
15 views

Directional Derivative in the direction in which $z$ is growing

Find the directional derivative of $f(x, y, z) = xy + 2xz - y^2 + z^2$, at the point $P = (1, -2, 1)$, passing through the curve $x = t, y = t -3, z = t^2$, in the direction in which $z$ is growing. ...
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1answer
27 views

Deriving the scalar/dot product without using the dot product itself?

How do you derive the result a.b = |a||b|cosθ I've seen some geometric derivations using the cosine rule but they then use the definition of a dot product to solve that but I wondered what a full ...
0
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1answer
53 views

What's the point of the cross product?

I don't understand the motivation behind defining cross products the way they're defined. Given two vectors $\vec{A}$ and $\vec{B}$ in $\mathbb{R^3}$, I can find a third vector $\vec{C}$ such that $\...