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.
11,980
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Orthogonal vectors in 4D
Consider two 4D vectors: $v_1=(\cos\varphi_1\sin\theta_1\sin\psi_1,\sin\varphi_1\sin\theta_1\sin\psi_1,\cos\theta_1\sin\psi_1,\cos\psi_1)$ and $v_2$, this vectors are orthogonal $v_1 \cdot v_2=0$, I ...
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Orthogonal projection of a vector on a subspace
I have to determine the orthogonal projection of the vector v=(2, 4, 2) on the subspace Span(v1, v2). I know that V=(v1, v2, v3) where v1=(1, 0, 1), v2=(5, 1, 1,) and v3=(4, -1, 0). So far I have ...
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Show that there exists a $λ ∈ K$ such that $l(A) = λtrace(A)$ for all $A ∈ Mat_{n,n}(K)$
Hey I want to check my solutions with this problem. Can someone help me?
Let $K$ be a field, $n ∈ \mathbb{N}$, and $l: Mat_{n,n}(K) → K$ a linear form such that $l(AB) = l(BA)$ for all $A, B ∈ Mat_{n,...
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In defining a directional derivative it is written $ \nabla f \cdot v=v f $, how is it? [duplicate]
while discussing 1-forms, it is given
Our goal is to generalize the concept of the gradient of a function to functions on arbitrary manifolds. What we will do is to make up, for each smooth function $...
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Apostol: How to calculate work by force field along intersection of sphere and cylinder.
This question is about line integrals in cartesian and cylindrical coordinates.
It is based on the following problem from Apostol's Calculus, Volume II, chapter 10 "Line Integrals", section ...
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Can a purely mathematical version of the right hand rule be given?
The right hand rule is a common convention for describing orientation of coordinates, used throughout physics. It's also used in the definition of the cross product.
Is it possible to give a purely ...
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Find the linear transformation in terms of the canonical base
Q. Consider the following direct sum decomposition of subspaces of $\mathbb{C}^4 = V_1 \oplus V_2\oplus V_3$. The subspaces are
defined by the canonical basis of $\mathbb{C}^4$ by the expressions $V_1 ...
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Spanning Set of a Subspace in $\mathbb R^3$
Find a spanning set for the subspace of $\mathbb R^3$ denoted by $U=${$(x_1,x_2,x_3) \in \mathbb R^3 | 3x_1 - x_2 -2x_3 = 0)$}.
My answer is span{$(1,1,1),(\frac{1}{3},3,-1)$}. Am I correct in ...
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Trying to understand the meaning of the following sentence from a research paper based on SVM classification
I am trying to understand the meaning of the following sentence given in a research paper based on SVM classifier.
Once all $\alpha_{\ell}$ values are obtained, we build a learning system which is ...
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What type of product should be applied to vectors with brackets?
When presented with a vector equation written as (a . b)c
How is c applied to the brackets? Is it dot product or cross product?
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Linear algebra. Finding three matrices in Vectorspace $V$
I have this vector space
$$V = \left\{\begin{bmatrix}a & b \\ c & -a\end{bmatrix}\, \middle|\ a, b, c \in \mathbb{R}\right\}$$
and am supposed to find three matrices ($A$, $B$ and $C$) in it ...
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How to find an equation of the plane
Find an equation of the plane passing through the point $P(1, 0, 1)$ and containing the line $[x, y, z] = [0, 1, -1] +t [2, 0, 1]$
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How to show that a map is $K$-multilinear
Hey I have this exercise where I have some questions
For a field $K$, let $x =\begin{pmatrix}x_1\\x_2\\x_3\end{pmatrix}, y = \begin{pmatrix}y_1\\y_2\\y_3\end{pmatrix}, z = \begin{pmatrix}z_1\\z_2\\z_3\...
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Finding parametric form of a circle that is in a plane $x-2y+z=0$ with radius 1 and center at origin
Find a parametric form of a circle that is in a plane $x-2y+z=0$ with radius 1 and center at origin
In the book they picked $\hat u =(2,1,0)$ (hat for unit vector) , did cross product with the plane ...
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How do you find the components of another vector given the components of a known vector and the angle between them in 3D?
I am working on a crystallographic problem. I am creating a 2D cross-section or slice of a surface in particular crystallographic directions. (I have found the angles of the extrema with respect to ...
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What should be the result of this Roto-translation? [closed]
Suppose I have a rotation matrix
mat = | 1 0 0 |
| 0 1 0 |
| 0 0 1 |
a translation vector
vec = | 1 1 1 |
and, I ...
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$u_1, u_2$ is linearly independent if and only if the family $u_1 + u_2, u_1 − u_2$ is linearly independent.
I have this exercise where I want to check my solutions. Can someone help me?
Let $u_1$ and $u_2$ be elements of $V$. If $1 + 1 ≠ 0$ in $K$, then the family $u_1, u_2$ is linearly independent if and ...
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Show that there exists exactly one $l_A$-invariant $K$-subspace of dimension $1$ in $K^2$.
So I have this exercise where I want to check my solutions. Can someone help me?
Let $A = \begin{pmatrix} a &b \\ 0& d \end{pmatrix} ∈ Mat_{2,2}(K)$ with $a, b, d ∈ K$, where $a ≠ 0$ and $d ≠ ...
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Find dot product of vector with tangent planes of a sphere using polar coordinates
Cartesian coordinates shown here are represented with
$$P(x, y, z)$$
while polar coordinates are to be in the form of
$$P(r; \theta; \phi)$$
As shown in the figure below. Note the use of semicolons to ...
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Step by step derivation of vector cross-product expression
Equation 1
$$\vec{a} * \vec{b} = (|\vec{a}| | \vec{b}| sin θ)ň $$
Equation 2
$$\vec{a} * \vec{b} = (a_yb_z - a_zb_y)î - (a_xb_z - a_zb_x)ĵ + (a_xb_y - a_yb_x)k̂ $$
I have managed to figure out and ...
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$Ax = y$ and $A'x = y'$ have the same solution set $L$, then $L = ∅$ or $\ker(l_A) = \ker(l_{A'} )$
Hey I have problems solving this exercise. Can someone help me?
Let $A, A' ∈ Mat_{m,n}(K)$ and $y, y' ∈ K^m$.
Let $l_A: K^n → K^m$ and $l_{A'} : K^n → K^m$ be the $K$-linear maps associated with $A$ ...
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Representing matrix of $id_V , f$ and $f^2$
Hey I have this exercise and I want to see if what I have done is correct.
Let $v_1$ and $v_2$ be a basis of the vector space $V$. Let $f: V \rightarrow V$ be the $K$-linear function with $f(v_1) = ...
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How to check whether a set is a subspace of a vector space?
Three vector spaces are given, and each one has a set that is potentially the subspace of the vector space.
$$
(1) \quad V = \mathbb{R}^4 \quad W=\begin{Bmatrix} \begin{bmatrix} a \\ -b \\ 2a+b \\ a \...
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Is there a name for the number of ones in a binary vector?
I'm creating two types of binary vector. The vectors have $M$ components and either $N$ ones or $(M-N)$ ones.
If you were to take all permutations of these vector types there would be a symmetry in ...
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Vector inner dot
Hi, I am reading Pure mathematics book, Volume one, chapter 11. It starts by introducing the inner product of vectors which it defines as: $\boldsymbol{A} \boldsymbol{P} = |A||P|\cos(\text{angle})$, ...
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local Coordinate transformation to global coordinates
Reference image
Please help me to prove;
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Write down the vectors - Please help me to solve it [closed]
A pipeline runs from a production plant ($P$) to the city of Cologne ($C$) which is $4\ \mathrm{km}$ east and $8\ \mathrm{km}$ north of the production plant. A second production plant ($Q$) is located ...
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Choose $r_i,b,z$ such that $\frac{\|x_1-y\|^2}{\sigma_1^2}+\frac{\|x_2-y\|^2}{\sigma_2^2}=\left\|\frac{x_1}{r_1}-\frac{x_2}{r_2}\right\|^2+b\|y-z\|^2$
Given $x_1,x_2,y\in\mathbb R^d$, I want to write $$\frac{\|x_1-y\|^2}{\sigma_1^2}+\frac{\|x_2-y\|^2}{\sigma_2^2}\tag1$$ in the form $$\left\|\frac{x_1}{r_1}-\frac{x_2}{r_2}\right\|^2+b\|y-z\|^2\tag2$$ ...
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What is the center vector of an open 2D cone
I have an open 2D Cone whose coordinates of vertex are $\{x,y\}$ and the half-angle of the cone is $\frac{\theta}{2}$, how can one find the center vector of this cone?
Any help is appreciated
Edit:
I ...
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How can I obtain a value from this formula??
I want to to know how to get a value using this formula if I have the values of:
$x_i , x_{i-1} , x_{i-2 }, y_i , y_{i-1} , y_{i-2}, z_i , z_{i-1} , z_{i-2}$
sorry i need 10 reputation for images,(...
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Understanding the trigonometry behind a pole leaning on a wall
I am copying and pasting this question I asked in the physics stack exchange. The reason I am doing so is because the question was automatically closed for being too "homework-like" and not ...
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Finding cartesian expression of vectors on an Equilateral triangle
Given an Equilateral triangle on the $xy$ plane with sides $a$, we pick the point $(0,0)$ to be at the center of the triangle(where the medians meet) and the direction of the $x$ axis to be parallel ...
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Vector related question
If $\vec{a}$, $\vec{b}$, $\vec{c}$, and $\vec{d}$ are unit vectors such that $$\Large(\normalsize\vec{a}\times\vec{b}\Large)\normalsize\cdot\Large(\normalsize\vec{c}\times\vec{d}\Large)\normalsize=1$$ ...
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$U$ be the set of all sequences of real numbers $(x_n)_{n\in \mathbb{N} }$ with $x_n + x_{n+1} = x_{n+2}$ find $\dim(U)$ and find a basis
Let $F = Abb(\mathbb{N}, \mathbb{R})$ be the $\mathbb{R}$-vector space of sequences of real numbers.
Let $U$ be the set of all sequences of real numbers $(x_n)_{n\in \mathbb{N} }$ with the property
$...
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Working with angles and Magnitude in vector geometry [closed]
This question for part B gives you 2 magnitudes and the angle between a vector and wants you to find the angle between the other two vectors. I was initially thinking of using the scalar product but i ...
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Give a complement of $U \cap V$ in $U + V$
Hey I have the following exercise where I am having a problem. Can someone help me?
Consider the following vectors in the$ \mathbb{R}$ vector space $\mathbb{R}^3$: $v_1 =\begin{pmatrix}1\\1\\2\end{...
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Exercise on a linear map
Hey I have a problem with this exercise. Can someone help me?
The erxercise is to show that there is a $\mathbb{Q}$-linear map $f : \mathbb{Q}^2 → \mathbb{Q}^2$ with $f( \begin{pmatrix}
1\\
0
\end{...
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Does there exist a $\mathbb{R}$-linear function $f : \mathbb{R}^3 → \mathbb{R}^2$ with $f(1; 0; 3) = (1;1)$ and $f(−2; 0; −6) = (2; 1)$?
Hey I have this problem where I want to see if my solutions are right.
Does there exist a $\mathbb{R}$-linear function $f : \mathbb{R}^3 → \mathbb{R}^2$ with $f(1; 0; 3) = (1;1)$ and $f(−2; 0; −6) = (...
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Some exercises about vector spaces
Hey I want ot check if my solutions for this exercise are right. Can someone help me?
Let $V$ be a finite dimensional $K$-vector space and $U_1, . . . , U_n$ a family of $K$-subspaces in $V$ .
Show ...
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Given 2 vectors in 3d find a condition that satisfies an inequality.
given $u=[1, 0, 1]$ and $v=[1, 2, -1]$. Let $x = sv + t\sqrt{2} \ u \times v$, where $s ∈ R$ and $t ∈ R$. Find the condition on $s$ and $t$ that renders $||x|| \leq \sqrt6$. Answer should be a ...
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How to find the distance before two points in a 2D plane?
I have the following scenario:
There's two different points in a 2D plane;
Both points are circles and haves X and Y coordinates and radius;
The X and Y coordinates of the points refers to the middle ...
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Relative Velocity Obstacles
I am reading Reciprocal Velocity Obstacles for Real-Time, Multi-Agent navigation and I have trouble to understand the main part of the paper:
Let's take an example,...
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How can I calculate a rotation matrix around an arbitrary axis?
Suppose I have a center at $(x_0, y_0, z_0)$, a vector $V$ goes from $(x_b, y_b, z_b)$ to $(x_e, y_e, z_e)$, and the translation vector is $(x_t, y_t, z_t)$.
I want to find the rotation matrix to ...
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Minimize sum of distance between random vectors - REFERENCES
I need some references (books, papers, applications, etc) of the following optimization problem:
Consider a set of reference random vectors
$\{{\bf x}_1,...,{\bf x}_p\}$, and a set of observed random ...
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1
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The sum of the angle between two vectors and their intermediate vector is greater than the total angle
I have two vectors $\vec{A}(2.735, -4.737)$ and $\vec{B}(2.735, 4.737)$, and a vector $\vec{P}(1.380, -0.796)$ in between these two vectors. When I calculate the angle between $\vec{A}$ and $\vec{B}$ ...
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Find a basis, and determine the dimension, for the space of all 2 × 2 matrices
So I understand how to find the basis and dimensions for linearly independent matrices, however this matrix is linearly dependent, I tried to look if any of the matrices can be created using the other ...
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Identifying level curves which are perpendicular to a particular vector field
In this problem I am given numerous vector field equations (both in 2 and 3 dimensions) and I am being asked which level curve/surface they are perpendicular to. The general process I've been using is ...
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Looking to Better Understand Conversions to Unit Vectors via dividing by magnitude
I understand that in order to scale a vector to its unit vector, we divide the vector by its magnitude (which is performing scalar multiplication to rescale the vector to having a magnitude of 1). I ...
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Linear Combinations: Is it possible for number of vectors to outnumber vector dimension?
In linear algebra. Just learned about linear combinations, and solved the following using gaussian elimination:
$$c1\begin{bmatrix} 1 \\\ -2 \\\ -5 \end{bmatrix} + c2\begin{bmatrix} 2 \\\ 5 \\\ 6 \...
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Geometrical area relationship using vector cross product
In order to show the result, the textbook quotes that:
Area of parallelogram ABEF = Area of parallelogram ABCD + Area of CDEF + Area of BCF - Area of ADE
I cannot see why this relationship is true - ...
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