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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1answer
21 views

Finding vertical projection of $\vec{a}$ on $\vec{b}$

I want to find the vertical or you could say perpendicular component of $\vec{a}$ on $\vec{b}$ Now I know that it can be found out using $\vec{a} - \left(\frac{a\cdot b}{|b|}\right)\vec{b}$ However I ...
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16 views

Derivatives of Matrices, Vectors and Scalar Forms that I can't find in Matrix Cookbook

Does anyone know what the derivative of $\frac{\partial (\bf{x}-\bf{\mu}_x)^T \Sigma(\bf{y}-\bf{\mu}_y)}{\partial \bf{x}}$, where $\bf{x}$, $\bf{y}$, $\bf{\mu}_x$ and $\bf{\mu}_y$ are vectors and $\...
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7 views

How to solve a L2 norm minimization problem with matrix orthogonal constraints

Given vectors $\mathbf{x}\in \mathbb{R}^N$ and $\mathbf{y}\in \mathbb{R}^M$ with $0\lt M\ll N$, I want to get an orthogonal $\mathbf{\Phi}\in \mathbb{R}^{M\times N}$ satisfying $\mathbf{y}=\mathbf{\...
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19 views

Proof of Vector Triple Product by Directions and Magnitudes

I'm trying to prove the vector triple product expansion by magnitude and direction: $$ \vec{a} \times(\vec{b}\times \vec{c})=(\vec a\cdot \vec{c})\vec{b} - (\vec{a} \cdot \vec{b})\vec{c} $$ The ...
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1answer
22 views

Find the resultant force

enter image description hereFind the resultant force when these three forces are applied to point A. I have tried to combine Force 1 and Force 2 together first, and used the resultant force to combine ...
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2answers
62 views

Find set of vectors orthogonal to $\begin{bmatrix} 1 \\ 1 \\ 1 \\ \end{bmatrix}$

$\mathbf{Question:}$ Find set of vectors orthogonal to $\begin{bmatrix} 1 \\ 1 \\ 1 \\ \end{bmatrix}$ $\mathbf{My\ attempt:}$ The vector is in $R^3$ so we can let vector $\begin{bmatrix} x_1 \\ x_2 \\ ...
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1answer
8 views

Calculated the distance between vector-line-equation and point. How do I find this point here?

I'm sorry for the weird title! I have a problem: Given is a point $p=\begin{pmatrix} 2\\ 2\\ 3 \end{pmatrix}$ and given is a line: $l(t)=\begin{pmatrix} 3\\ 3\\ 6 \end{pmatrix}+t \begin{pmatrix} 1\...
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34 views

Locus of points r such that r.n=d where d is a constant

The Question Let n be a unit vector, and let d be a constant. What is the locus of points r such that $\mathbf{r}.\mathbf{n}=$d ? The Attempt So I know that $\mathbf{a}.\mathbf{b}=\left|a\right|\left|...
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1answer
16 views

Convective derivative vs Divergence of velocity

What is the physical significance or difference between Convective derivative : $\vec{v} \cdot \nabla $ and the Divergence of velocity $\nabla \cdot \vec{v}$? I have understood the convective ...
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28 views

Confusion about derivative layouts and shapes

This is related to a question I asked earlier. If $x\in R^n$, $f(x)\in R^m$. What would $\frac{df(x)^T}{dx}$ be? In the book I'm reading it says that numerical layout of derivatives has been used in ...
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35 views

Confused about how we scale graph axis' to make the axis' dimensionless.

I am trying to understand the solution to part $\mathrm{(iii)}$. But, for the question I'm asking to make sense I need to include the solutions to parts $\mathrm{(i)}$ and $\mathrm{(ii)}$ also: ...
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5 views

Covariance and Contra-variance w.r.t. row column and inverse change of basis matrix

Could someone explain in an intuitive and if possible as simple as possible manner (ideally with at least 1 example) the relationship between covariance/contra-variance and row/col vectors ? E.g. ...
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2answers
43 views

Vector equivalent of complex multiplication and division

I understand that the addition and subtraction of complex number is the same as vector addition and subtraction. But what is the vector equivalent of multiplication and division of complex numbers?
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24 views

Decompose 3d vector into 3 vectors with equal (or other ratio) length, each in a different ortogonal plane

I'm trying to solve a problem with a mechanical (actually game) application. I have a vector/normal in 3 dimensions (so with x,y,z). I want to decompose this vector into 3 vectors, where each vector ...
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57 views

Tensor algebra and calculus [closed]

The $i$−th component of the cross vector product can be written as $t = b \times c$, $~t_i = \varepsilon_{ijk}b_jc_k$. Use this to show that $(a, b \times c) = (c, a \times b) = (b, c \times a)$. how ...
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7 views

Pertubation stream function [closed]

Why are the components of the perturbation stream function of the opposite sign to a standard stream function? Any help would be much appreciated, thanks.
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1answer
20 views

How compute the triple product having only the module of vectors

In this case I have this informations $|\vec{u}|=3, |\vec{v}|=\sqrt{3}$, and $\vec{u}\cdot\vec{v}=3$. But I don't know how I can find the components $\vec{i}, \vec{j}$ and $\vec{k}$ to calculate the ...
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109 views

Calculate 3D Rectangle from 4 projected points on screen

Given 4 known projected points on the screen, I need to calculate a 3D rectangle where the 4 projected points coincide with the rectangle corners from the o ...
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2answers
50 views

How do you find the angle between two vectors? [closed]

I am asked to find the angle between the lines $$\frac{x-1}{2}=1-y=2z \text{ and } x=y=3z,$$ but I don't know how to manipulate the negative $y.$ I found the dot product of the two direction vectors $...
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2answers
33 views

Geometrical explanation / visualisation as to why 2 vectors cannot span R3?

I have already understood as to why 2 vectors cannot span R3 in a simple algebraic manner. This is from an answer to this very question 3 years ago : Why two vectors cannot span ${\bf R}^3$? However, ...
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1answer
15 views

To find the angle a particle makes with the horizontal at any time 't'

Should you vector sum the position vectors at this time 't' or vector sum the velocity vector at this time 't' to find the angle a particle makes with the horizontal at any time 't'
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30 views

Vector methods to find line of intersection of two planes

If we are given two non-parallel planes $A$ and $B$ where $\textbf{a}$ and $\textbf{b}$ are unit normal vectors to $A$ and $B$ respectively, and we are also given that the points with position vectors ...
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0answers
61 views

The convex hull of a vector space.

For a matrix $A = (a_1, \ldots, a_n) \in \mathbb{R}^{m \times n}$, denote \begin{equation*} \| A\|_{b, c} := \big\| (\| a_1\|_b, \ldots, \| a_n\|_b)^{\top} \big\|_c. \end{equation*} where $\| \...
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0answers
11 views

Does A vector-valued random variable X = [X1…Xn]^T mean that any Xn is a vector?

Does the quote "A vector-valued random variable X = [X1....Xn]^T is said to have a multivariate normal (or Gaussian) distribution..." from image attached mean that each X1 is a vector of ...
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1answer
27 views

Check whether the set $S$ of all vectors in $\mathbb R^3$ given by $S=\{(a + 2b, a +3, b):a, b \in \Bbb R\}$ is a subspace of $\mathbb R^3$.

The set $S$ of all vectors in $\Bbb R^3$ given by $S=\{(a + 2b, a +3, b):a, b \in \Bbb R\}$ is a subspace of $\Bbb R^3$. Justify your answers
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3answers
35 views

Given two defined points A and B, what points X exist such that the vectors AX and BX are orthogonal?

For class I have to find: Given two points, say A (1, 1, 1) and B (5, 5 ,5), what is the set of points X (x, y, z) that exist such that the vectors AX and BX are orthogonal? I know that the points X = ...
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3answers
59 views

The point $O$ placed inside triangle so that $\vec{OA}+2\vec{OB}+3\vec{OC}=0$

From the triangle $\triangle ABC$ we have $AB=3$, $BC=5$, $AC=7$. If the point $O$ placed inside the triangle $\triangle ABC$ so that $\vec{OA}+2\vec{OB}+3\vec{OC}=0$ , then what is the ratio of the ...
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0answers
18 views

Proof linear dependence if one of the vectors is a linear combination [closed]

Prove that the set of vectors p1,p2,p3...pn is linearly dependent if and only if one of the vectors is linear combination of the rest
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84 views

Why does a spiral structure appear for this vector field?

Consider the following 2D vector field on the $xy$-plane $$\vec{V}=\begin{pmatrix} -(m^2-md+x^2)\cos{2d\,t}+xy\sin{2d\,t} \\ -xy\cos{2d\,t}+(m^2-md+y^2)\sin{2d\,t} \end{pmatrix}$$ where $d=\sqrt{x^2+y^...
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1answer
12 views

Let a,b,c be distinct non-negative numbers. If the vectors ai+aj+ck,i+k and ci+cj+bk lie in a plane, then relation between a, b and c

My Question is How to proceed with the question. I mean there are two ways:- Either one could apply the scalar triple product and solve the determinant equated to 0 which has provided me with the ...
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1answer
27 views

Let P and Q are two points on curve $y=\log_{1/2}(x-\frac 12) + \log_2(\sqrt{4x^2 -4x +1})$ and P is also on $x^2+y^2=10$..

Let P and Q are two points on curve $y=\log_{1/2}(x-\frac 12) + \log_2(\sqrt{4x^2 -4x +1})$ and P is also on $x^2+y^2=10$. Q lies in the given circle such that abscissa is an integer. Find the ...
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1answer
59 views

What is a Surface

Let $f$ be a differentiable function and $c$ a number, The set of points $P$ s.t $f(P)=c$ and $\nabla f(P)\ne 0$, is called a surface. (The author didn’t give any informations about the domain and the ...
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1answer
25 views

Transformation matrix for rotation by angle theta [closed]

Find the standard matrix of the transformation T: R3 → R3 that corresponds to the anti-clockwise rotation by an angle θ about the x1-axis. I have no idea how to even begin to approach this problem. ...
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0answers
20 views

Vectors and polar coordinates [closed]

First part Second Part I understand parts i and ii. I am confused on parts 3 through 6.
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39 views

Matrix norm ineequality

Let $ A= \begin{bmatrix} a & b \\ c & -a \end{bmatrix}$ with $a, b, c ~$ are complex nubers satisfying $0\leq a <1$ and $\vert b \vert=1$. Prouve that $$ a^{2}+1\leq \Vert A \Vert .$$ I ...
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0answers
16 views

i have a set of vectors and i have to check if the sentence is true or not [closed]

i have a question that goes this way i am given S=(v1,v2,v3) and S is group of 3 vectors From vector space V and this question is if S is Linear dependent then v3 belongs to span{v1,v2} if dim(V)=3 ...
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2answers
108 views

Prove $P^k = P$ for all $k=1,2,\ldots$ [closed]

Let $P = A(A^TA)^{-1}A^T$, where $A$ is an $m \times n$ matrix of rank $n$. Prove that $P^k = P$ for all $k=1,2,\ldots$
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1answer
19 views

Orthonormalized basis of linearly dependent system

Given the following vectors: How is the orthonormalised basis of the span{v1, v2, v3} calculated? Given it is linearly dependent do you just need to work out the orthonormalised basis of the span {v1,...
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2answers
42 views

Proving that $\Vert Ax\Vert \leq \Vert A\Vert \Vert x\Vert$ where $A$ is a matrix and $x$ a vector

I would like to prove that $\Vert Ax\Vert \leq \Vert A\Vert \Vert x\Vert$ where $A$ is an $n\times n$-matrix and $x = \left(\begin{array}{c} x_1\\ \vdots\\ x_n\end{array}\right)$ is a vector. I've ...
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0answers
27 views

how can I prove $(A\cdot\nabla)A=\frac12\nabla A^2-A×(\nabla×A)$? [closed]

$$(A\cdot\nabla)A=\frac12\nabla A^2-A×(\nabla×A)$$ help me prove this So i try to proof this equation. We're talking about vectors, divergence, and curl. But i dont really know how come the left side ...
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1answer
28 views

Shortest distance between two particles whose position vectors depend on time

The Question At time $t$, two particles have position vectors given by r$_1(t)$ and r$_2(t)$ where $$\mathbf{r}_1=\mathbf{a}_1+\mathbf{b}_1\left(t-t_1\right)$$ $$\mathbf{r}_2=\mathbf{a}_2+\mathbf{b}_2\...
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1answer
40 views

plotting 2 x 2 matrix visually. What it represents?

Hello I am engineering student. I need help understanding basic of matrix plotting. Currently I am learning about vectors and I wonder how they are represented by a matrix. Suppose we have a 2D vector ...
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1answer
41 views

Why is the term “$1 \times k$ vector” used rather than $k$-length vector?

I see well informed people using the term "$1 \times k$ vector" in lectures. My understanding is that a vector is necessarily $1 \times k$, such that the "$1$ times" part is ...
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0answers
12 views

How can I analyse a vector transformed by a special symmetric matrix?

Given a vector $\mathbf{x} \in \mathbb{R}^{N}$, a matrix $\mathbf{A}\in \mathbb{R}^{M\times N} (M\ll N)$, and a constant positive real number $\rho$, my target vector $\mathbf{y}\in \mathbb{R}^{N}$ is ...
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2answers
31 views

Finding the tangent to a parametric curve $(t^3, t^5)$ at $(0,0)$

The curve $(t^3, t^5)$ at that point $(0,0)$ does not have a tangent vector as when you work it out, you will arrive at $(0,0)$. Question: How can you find a new parametrisation for the curve such ...
3
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3answers
60 views

Finding set of vectors satisfying $\textbf r+\textbf r\times\textbf d=\textbf c$

Let $\mathbf{c}$ and $\mathbf{d}$ be fixed vectors in $\mathbb{R^3}$. Find all vectors $\textbf r$ such that $\textbf r+\textbf r\times\textbf d=\textbf c$. Attempt: Take the vector product with $\...
2
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1answer
47 views

The angle between two vectors

I'm just a new person here and hope that I can ask my question properly. Here is my question: Find the vector $\vec{b}$, when the angle between two vectors is $30^\circ$ and one vector is given by $\...
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1answer
48 views

Solving vector calculation question

A spaceship is traveling through space at constant speed along a straight line that passes through the points $A = (-3,-8,-6)$ and $B = (-8,-4,-3)$. The star Gliese $061$, located at $P = (-12,0,0)$ ...
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0answers
11 views

With Geogebra, how to defne a position vector in a moving frame $(O', \vec{i'},\vec{j'})$ in order to have this vector placed at $O'$,not at$O=(0,0)$.

Context of my question : understanding some basic cinematic equations by creating a toy model of moving 2D relative referential in Geogebra . In Geogebra, it's possible to create a relative 2D ...

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