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.
12,707 questions
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What is a good notation for a shifted (rotated?) vector?
Suppose I have a vector $X$. What would be a good way to denote the vector $Y$, with $Y_1=X_N$ and $Y_{i+1}=X_i$? Is there a standard?
I'm looking for a possible convention like $X^\text{SH}$ or $X^*$ ...
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What is a good notation for a randomly sorted vector?
Suppose I have a vector $X$. What would be a good way to denote the vector $Y$, where each element $Y_i$ is randomly picked without replacement from $X$? Something like $X^\text{RS}$ or whatever. ...
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Vector Rotation Communative Property [closed]
I understand 3D object rotation is non-commutative, however a 3D vector has 2 axes of rotation, not 3. Is a 3D vector's rotation commutative?
3D Vector: <x,y,z>
Intuitively, it only needs to ...
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Finding the shortest distance and time for it using relative velocity
Two airplanes A and B flies with constant velocities in the same height as 300 km/h to the direction which makes an angle $\alpha$ to the west from north and 400 km/h to the west respectively, here $\...
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Finding the accelerations and tension of a pully wedge system [closed]
According to the figure below, a wooden cuboid A of mass "M" is suspended to a horizontal fixed straight inextensible string which passes through 2 smooth rings which are attached to the ...
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1
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2
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Why are the projections of an area element onto the coordinate planes the components of its normal vector? [duplicate]
I understand that the components of the cross product $\bf{u}\times\bf{v}$ are the area of the parallelogram enclosed by $\bf{u}$ and $\bf{v}$, projected onto the respective coordinate planes (e.g. ...
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XYZ coordinate distance
I was presented with this question...
So a plane that traveled 1 mile at an incline of $45^\circ$ from the ground $(Z)$ and angle of $30^\circ$ towards the North $(Y)$ from the East $(X)$, went a ...
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Finding dimension of subspace of a vector space [closed]
Let $S$ be a set of vectors in $\mathbb{P}$$_2$ which are of the form $ax^2+bx+c$ where $a,b$ are scalars.Then what is the dimension of $S$?
If $c$ is a polynomial function of $b$ passing through ...
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Plane/line distance problem: finding a point given equations of plane and line
Here is the question:
The point $Q$ has position vector $(7,4,6)$, the plane $P$ has equation $2x + y + 3z = 36$, and the line $L$ has equation $(20, -8, 1) + t(-7, 5, 3)$.
$Q$ lies in the plane $P$. ...
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Prove that if a subspace $S$ of $\mathbb R^n$ contains a strongly positive vector, then $S$ has a basis of strongly positive vectors.
A vector $v= (a_1,...,a_n) \in \mathbb R^n$ is called strongly positive if $a_i>0$ for all # $i=1,...,n $
a) Suppose that is $v$ is strongly positive. Show that any vector that is “close enough” to ...
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Rotating A Vector Keeping Angle From Origin Unchanged [closed]
I am looking to rotate a vector in a way which does not change the relative distance or angle from origin. This is to ensure that the distance of any random vector is the same from the original vector ...
1
vote
1
answer
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Prove that: $a\cdot IA^2+b\cdot IB^2+c\cdot IC^2=abc$. [closed]
Given triangle $ABC$ with $BC=a, CA=b, AB=c$,
let $I$ be the center of the circle inscribed in the triangle.
Prove that: $a\cdot IA^2+b\cdot IB^2+c\cdot IC^2=abc$.
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Why is $dA_{\perp} = dA\cos(\theta)$?
I'm sure there's an easy fix to this, but I can't seem to find it. Let $dA$ be a tiny patch of area on an arbitrary surface. Let $dA_{\perp}$ be the part of this area that is parallel to the vector ...
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Why vectors multiplication is their dot-product [closed]
I have an expression:
$$|\vec{v} - \vec{w}|^2$$
I know I can simply substract the vector $\vec{v}$ from the vector $\vec{w}$ and then calculate resultant vector magnitude, but I want to do that by ...
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Looking for general advice for finding collection of evenly spaced vectors as determined by absolute cosine similarity
I have a problem where I'm trying to find a collection of $N$ unit vectors, $V = \left\{ \hat{v}_1,\hat{v}_2,...,\hat{v}_N\right\}$, on the half-sphere in $\mathbb{R}^3$ such that such for $\hat{v}_i=(...
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But why is the curl of a vector field a cross product?
I'm a first year undergrad student who is learning vector calculus. I've reached out to my professor at university, read math textbooks and browsed several online resources, but one thing has been ...
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2
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How do I rotate one direction vector by another vector's direction away from positive z? [closed]
Let's say I have two normalized (-1 to 1) direction vectors A and B:
...
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1
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Simple relationship between correlations
The following question rose from work with time-series but I generalized it so that it might be interesting to a larger audience:
Let A, C be vectors of $\mathbb{R}^n$ and B,D vectors of $\mathbb{R}^...
4
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2
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If $\operatorname{dim}(V)=n$, then $\operatorname{dim}(\wedge^n (V))=1$
Context
Suppose $V$ is a $n$-dimensional vector space over $\mathbb{R}$. In our differential geometry class, we define the tensor product
\begin{align}
\overbrace{V\otimes \cdots \otimes V}^{k\text{ ...
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1
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How to find a plane through a point and a line without using vectors? [closed]
Given the plane $\pi$ represented by $2x + 3y - z + 5 = 0$ and a point $P(-1, -1, 2)$:
I need to verify whether the point $P$ belongs to the plane $\pi$;
I need to determine the plane $\pi'$, passing ...
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What is the expected overlap between two random unit complex vectors of length $L$?
Given two random, uniformly sampled unit vectors $v_1, v_2 \in \mathbb{C}^L$, what is the expected value of their overlap $\langle v_1, v_2 \rangle$? I need a formula to calculate this value, given ...
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$S^3$ to describe three-vectors
In this wiki article about topological defects ( see the "To restate more plainly..." paragraph is remarked that a $3$- sphere $S^3$ can be used to encode a three-vector, its direction plus ...
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1
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87
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Use determinant to compute the volume of a "box" with vertices $(0,0,0), (3,1,1), (1,3,1), (1,1,3)$
So I graphed the vectors and it formed a pyramid instead of a "box":
I know that the volume of a "box" or parallelepiped is the triple product or the determinant of a $3 \times 3$ ...
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1
answer
31
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Why two vectors in the same coset must have same distance to a lattice
I'm trying to understand the post-quantum cryptography standards released by NIST, which led me to learn about lattices. My knowledge of group theory isn't great but I followed along with the basics. ...
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2
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70
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Flux of $ (-\sin y)\mathbf{i} + (x \cos y)\mathbf{j}$
I am trying to calculate the outward flux of the vector field $\mathbf{F} = (-\sin y)\mathbf{i} + (x \cos y)\mathbf{j}$ across the square in the first quadrant bounded by $0 \leq x \leq \frac{\pi}{2}$ ...
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The meaning of the formula finding the area of a triangle in coordinates
I was trying to find out the area of a triangle given three points in the coordinates (I didn’t search on the internet, I later found out that this was already done by someone). I got the formula as ...
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Determining if a set of vectors is a basis
Given a set of vectors in some space $\mathbb{R}^{3}$ where
$$V = \{v_{1}, v_{2}, v_{3}\} \in \mathbb{R}^{3}$$
and $v_{n}$, $n=1,2,3$ is a column vector.
And we have a matrix $$A=(v_{1}, v_{2}, v_{3})$...
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2
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What exactly is the geometrical meaning of a dot product? Why does it use both vectors instead of just the projection of one vector onto the other? [duplicate]
I'm just starting to learn about vectors, and I am a bit confused. According to my reference material, a dot product is the degree of "alignment" between two vectors.
Consider two vectors a ...
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when to take express rows in matrix, and when as columns.
Usually in my class, we express them as columns of the matrix but only for some questions are the vectors taken as rows, but even then the method for calculating row and column spaces didn't change. ...
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4
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The Dot Product of Position Vector $\vec{r}(t)$ and Velocity Vector $\frac{d\vec{r}(t)}{dt}$
As excerpted above, Wolfram Mathworld states this:
If $\vec{r}(t)$ is the position vector, then
$$ \vec{r}(t) \cdot \frac{d\vec{r}(t)}{dt} = |\vec{r}(t)| \left| \frac{d\vec{r}(t)}{dt} \right|.$$
I ...
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Find Vector From 3 points
How to find the unit vector $\vec{CD}$ perpendicular to line segment $\bar{AB}$ pictured here?
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The Linear (In)Dependence of Rows of $\mathbf{X}$ and (Non)Trivial Solutions to $\mathbf{Xa}=\mathbf{0}$.
Let $\mathbf{X}$ be an $m\times n$ matrix. Let $\mathbf{a}$ be an $n\times1$ vector. Consider the homogeneous equations:
$$
\mathbf{Xa}=\mathbf{0}.
$$
It is known that the columns of $\mathbf{X}$ are ...
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3
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Why can't (1,0) be the null vector for the vector space in which (1, x)+(1, y) = (1, x+y)?
I have been trying to see if...
v = { (1,x) ∈ R² }, (1, x) + (1, y) = (1, x+y), k·(1, x) = (1, k·x)
...is a real vector space or not. I can't understand why isn't (1,0) a valid null vector, since it ...
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1
answer
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determinant of $A = \begin{pmatrix} 0 & x^T \\ x & C \end{pmatrix}$, where $x$ is n-dimensional column vector and C is $n \times n$ dimensional matrix
I'm trying to find the determinant of
$
A = \begin{pmatrix}
0 & x^T \\
x & C
\end{pmatrix}
$, where $x$ is an n-dimensional column vector and C is an $n \times n$ dimensional square and ...
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Can an equation with a norm be put into purely linear and quadratic terms?
I am attempting to use least-squares methods to estimate parameters in a cone function. I have the following equation of a single sided cone in $\vec{x} \in \mathbb{R}^3 $:
\begin{equation}\large f(\...
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Why is the angular velocity differentiated as a constant?
Kinematics, uniform motion around a circle.
The angular velocity, which is defined as $\omega=\frac{\varphi}{t}$, when a point is rotated by an angle of $2\pi$ for a time equal to the rotation period, ...
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Integral over unit sphere of radius vector tensor field
How do I solve the following two integrals over the unit sphere?
$$\iint_S \vec{r_0}\otimes\vec{r_0}dS \\ \iint_S \vec{r_0}\otimes\vec{r_0}\otimes\vec{r_0}\otimes\vec{r_0}dS$$
I undestand I should use ...
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1
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Software for closest vector problem in $F_2$ [closed]
I need to solve exactly the closest vector problem over the field $F_2$ with 2 elements, and I am looking for a software that can do better than my brute-force code in C.
Specifically, I have a list ...
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Picking orientation of three vectors to minimize volume.
Assume you have three vectors in $\mathbb{R}^3$ Then they form an infinity pyramid with its apex at the origin (if we consider the lines spanned by the vectors).
This partitions the space into two ...
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0
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hyperplane intersecting a polyhedral cone
$\newcommand{\R}{\mathbb R}$
Consider a family $V$ of non-null vectors in $\R^n$. Consider the (open) polyhedral cone
$$C(V) = \{x\in\R^n\;:\;v\cdot x> 0\,,\;\forall v\in V\}\,.$$
We will call $C(V)...
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Computing the commutation relation in vector form
I am trying to prove the commutation relation:
$$[p^2,\frac{\vec{r}}{r}]=\frac{2}{r^2}(\hat{r}+i\hat{r}\times \vec{L}).$$
where $\vec{p}=-i\vec{\nabla}$, $\hat{r}=\frac{\vec{r}}{r}$ and $\vec{L}=\vec{...
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Why do we care about the binormal vector and osculating plane?
In Calculus 3 we learn about vector functions, and one of the first things we learn about them is how they can create space curves/paths. The first applications we learn about has to do with physics, ...
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0
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Correct notation for matrix and vector repetitions
Given a matrix $A^{w \times f}$. I repeat the matrix $L$ times along the $f$ dimension, resulting in $A'^{w \times Lf}$. What is the proper notation for the repetition operation?
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2
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finding the shortest distance using relative velocity
let v and r be the velocity and displacement vectors, at an instance considering 2 bodies A and B
$$r_A = 13i + 5j$$
$$r_B = 3i - 5j$$
$$v_A = 3i - 10j$$
$$v_B = 15i + 14j$$
the question asks, ...
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1
vote
1
answer
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Auto-Stereograms. Find vectors whose elements are limited to the range 0 to 1 and which satisfy the following equations
I figured out what I believe is an entirely new way of generating auto stereograms (3D without glasses). It relies on being able to solve the following equations. Given two sets $l_i$ and $r_i$, each/...
4
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0
answers
40
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Assigning mass equally around a wheel [closed]
I need to fit 25 similar parts around a wheel in 25 fixed, equally spaced locations. The parts are of similar but not equal mass - about 1% total variation. Is there a formula (or better still a ...
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1
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Commutativity of vector addition in the context of displacement addition
Why do we use addition of vectors as a mathematical model when adding displacements? One of the properties of this operation is commutativity. When we try to apply the commutative property to two ...
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1
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Calculating magnitude of basis vectors for polar coordinates
Let $M=\mathbb R^2\setminus\{0\}$ be a smooth manifold. Let $(U,\phi)$ be one of its coordinate charts where $U\subseteq M$ is open. Fix $p\in M$. Let $\{\frac{\partial}{\partial r}, \frac{\partial}{\...
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1
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Showing coplanar vectors are also colinear
I have a question regarding this problem: if vectors $\vec a \times \vec b,\vec b \times \vec c,\vec c \times \vec a$ are coplanar, then they are also colinear.
I visually understand this, but don't ...
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0
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find the position vector within a system of coplaner points
a,b,c,d are position vectors of four coplaner points A,B,C,D. Here $d = \lambda a + \mu b + \gamma c$ and $\lambda + \mu + \gamma = 1$. AB and CD intersects at E. how to show that the position vector ...
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