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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Given the direction cosines of two mutually perpendicular lines, show the direction cosines of the lines perpendicular to the above two lines are:

I just want to ask that if it is written two equations of relations of l, l1, m, etc. How did he write it in that ratio or fractional form like l upon something and m upon something etc and made it ...
seven65ive's user avatar
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Center of Mass and Concurrency of Lines

Given points $v_1$, $v_2$,...,$v_p \in \mathbb{R^n}$ and corresponding "masses" $m_1, m_2...,m_p>0,$ the center of mass can be defined as: $$\frac{m_1v_1+m_2v_2+...+m_pv_p}{m_1+m_2+...+...
Unknown's user avatar
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How to get a 3D vector in the same direction as another vector limited by an angle? [closed]

If I have two 3D vectors as shown by the blue and red vector in the image below, where the red vector is at an angle of more than 20 degrees from the blue vector, how could I calculate a new vector (...
Mori's user avatar
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4 votes
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102 views

Which is the correct definition of covectors?

Some says covectors are linear map that maps $ V \mapsto R $ (which means it's just a row vector considering vectors are $ n $x$ 1 $ matrix and mapping is matrix multiplication), while some say it's a ...
posfn0319's user avatar
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How would I solve for the s_\alpha or s_r vectors using components from the listed E-Frame? Space Vehicle Dynamics

I'm taking a dynamics class where we are specifically focusing on the transport theorem. I need to first identify the position vector, and then take 2 derivatives for inertial velocity and inertial ...
awesomejack02's user avatar
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41 views

Am I correct in terms of understanding gradient and covector?

Well actually I know what gradient is and what that means. My professor once said that gradient is actually a covector in passing. As I know, covector is a linear map that maps vector to a scalar but ...
posfn0319's user avatar
1 vote
2 answers
59 views

Prove $\frac{|a+b|}{|a-b|} = \cot(\frac{\alpha}{2})$ where $a$ and $b$ are vectors of equal magnitudes.

Prove $$ \frac{|a+b|}{|a-b|} = \cot\Bigl(\frac{\alpha}{2}\Bigr) $$ where $a$ and $b$ are vectors of equal magnitudes and $\alpha$ is the angle between $a$ and $b$. My approach: Let $$ \left\{ \begin{...
Crustocean 01's user avatar
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In the usual notation, let 𝑖 + 2𝑗 and 𝑖 βˆ’ 3𝑗 be the position vectors of two points 𝐴 and 𝐵, respectively,

In the usual notation, let $𝑖 + 2𝑗$ and $𝑖 βˆ’ 3𝑗$ be the position vectors of two points 𝐴 and 𝐡, respectively, with respect to a fixed origin 𝑂. Find the position vectors of the two distinct ...
marry's user avatar
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let 𝛼𝑖 + 𝑗 and 𝑖 βˆ’ 2𝛼𝑗 be the position vectors of two points 𝐴 and 𝐵

Let 𝛼 > 0 and in the usual notation, let 𝛼𝑖 + 𝑗 and 𝑖 βˆ’ 2𝛼𝑗 be the position vectors of two points 𝐴 and 𝐡, respectively, with respect to a fixed origin O. Also, let C be the point on AB ...
marry's user avatar
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Transform a vector of positive and negative values to sum up to 0

Is there a transformation that produces a vector with sum $0$? There are positive and negative values and the transformation does not need to be preserve the weights. E.g.: $f(x_1, x_2, x_3) = (x_1', ...
Philohippo's user avatar
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Need some help with vectors. [closed]

Given \begin{align} u &= βˆ’5i + 3j \\ v &= 2i + 5j + 4k \end{align} compute $u\times v$ $v\times u$
Ineedserioushelp's user avatar
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converting pose (which is a quaternion & a vector) from a coordinate system to another

The question is about a world and a camera that is defined in this world. I want to transform the pose (which is a rotation and a translation) of the camera given in the world coordinate system (...
Hana's user avatar
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How to generate (normally or uniformly distributed) random vectors (directions) on the restricted surface of a specific hyperplane

I would like to generate (normally and/or uniformly distributed) random vectors (directions) on the (restricted) surface of a given hyperplane. More specifically, in the two-dimensional space, the ...
Israfil Roshdi's user avatar
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How to create row/column selection matrix from a binary vector?

Given $n \leq m$, let $A$ be a matrix of shape $(n, m)$, and $x$ be a binary vector of length $m$ (it's guaranteed that there are exactly $n$ 1's in $x$). People could extract the columns of $A$ ...
Andy Liu's user avatar
3 votes
0 answers
91 views

What is the opposite of antiparallel?

Is parallel really the opposite of anti-parallel? While anti-parallel indicates that two vectors are directed along the same line and are pointing in opposite directions then it seems to me that ...
Steeven's user avatar
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2 votes
2 answers
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Showing that $\boldsymbol{\nabla}r^n=nr^{n-2}\mathbf{x}$

In three dimensions, use suffix notation and the summation convention to show that $$\boldsymbol{\nabla}r^n=nr^{n-2}\mathbf{x},$$ where $\mathbf{a}$ is any constant vector and $r=|\mathbf{x}|.$ My try:...
Bagaringa's user avatar
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Are the columnspace and the nullspace not always the same for some matrix A (mxn) and some product of matrices BA, where B is invertible (mxm)?

Why is this not always true? My rationale is that some vector c (mx1) must either be in the columnspace or nullspace of A. If c is in the nullspace of A, it must be in the nullspace of BA. If c is the ...
Questionasker's user avatar
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Determining if two particles collide in a 2d container

I'm trying to determine if two particles collide within a certain period of time in a 2D container. Given the positions and velocities of the particles, I know a formula to determine if there is a ...
Trom's user avatar
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2 votes
1 answer
72 views

orienting a point in polar coordinates along a particular unit vector

I have the center of a circle $\vec{c}$ in 3 space and the radius $r$. I also have a unit vector $\hat{v}$ defining the orientation of the plane of the circle. I wish to parameterize this circle and ...
Stan Shunpike's user avatar
-3 votes
1 answer
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4 elements are named x,y,z,w now how to name 8 elements [closed]

I have a vector of four elements. The elements are named x, y, z, w: Vec4(x, y, z, w) Now I have another vector of eight ...
Megidd's user avatar
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How to properly (rigorously?) treat a dot product integral?

I have a very long cylindrical mass. Now say I have some cylindrical Gaussian surface $\Omega$ centered around this cylinder, visualized in purple below (ignore everything else in the diagram), for ...
Max0815's user avatar
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2 votes
2 answers
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Perpendicular diagonals of quadrilateral $ABCD$ meet at $O$. Find the angle between lines $AB$ and $DC$, given lengths of $OA$, $OB$, $OC$, $OD$.

A few days ago I was at the math Olympiad and I failed to solve only one rather simple task, it feels like a 9th grade, but I either misunderstand something, or there is something quite interesting ...
NeizvesnNo's user avatar
2 votes
0 answers
117 views

Build a new linear mapping $\psi$ and the matrix of this mapping

Task: The linear mapping $\phi: \mathbb{R}^5 \rightarrow \mathbb{R}^5$ in a pair of standard bases has a linear mapping matrix A: \begin{equation*} A = \left( \begin{array}{ccccc} 2 & -2 & -3 &...
Little Mandelbrot's user avatar
1 vote
1 answer
67 views

geometric interpretation on covariant derivateve in curvilinear coordinates.

I'm having trouble understanding what does the covariant derivative do in a coordinate system where we have changing basis vectors. I always thought it was giving us the change in coordinates while ...
Krum Kutsarov's user avatar
1 vote
1 answer
32 views

Finding angle relative velocity makes with unit vector i.

I've attempted this question but can't seem to get the right answer. Three identical particles A, B and C are moving in a plane and, at time $t$, their position vectors, $Γ’$, $bΜ‚$ and $Δ‰$, with ...
Developer's user avatar
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What this formula for product of two complex vectors is?

I read in one russian book about complex vectors. There was formula for product of two complex vectors. I was confused because I didn't find what this formula is. $$ \begin{aligned} \boldsymbol{z}_{1} ...
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Why is $\textbf{A} = A_i \textbf{e}_i$ not independent of choice of coordinates?

I'm currently reading "Tensors, Differential forms and Variational Principles" By D. Lovelock and H. Rund. This is from page 2 : and then on page 7, concerning equation 1.1 I fail to ...
aidaGoG's user avatar
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Intuition behind pre-multiplication of a vector with a matrix

Given a matrix $\mathbf{A}$ and a vector $\mathbf{x}$, the MVP $\mathbf{Ax}$ gives the geometric interpretation of transforming $\mathbf{x}$ into the coordinate system with basis vectors defined by ...
Susmit Agrawal's user avatar
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Find all unit vectors u in the xy-plane that make an angle of 45 degrees with v = [1,0,1]

For this question, I started with the dot product of an arbitrary vector u and v: uβ€’v = ||u|| ||v|| cos(45) = (1) (√2) (√2/2) = 1 Then I used the u1v1 + ... + u3v3 ...
Hagar Ali's user avatar
1 vote
0 answers
25 views

Confusion about the order of rotations when using rotation matrixes

I am studying rotations matrixes and I'm confused about the order of rotations. Let's say I would like to rotate vector $\vec a$ into $\vec b$. They are defined in the cartesian coordinate system. I ...
Nikola Ristic's user avatar
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3 answers
113 views

Geometry problem: finding coordinates on outer circle using extrusion from centre through point of inner circle

My problem involves an inner circle (centre $I$) with a displaced placement within an outer circle (centre $O$). I have a point placed somewhere on the circumference of the inner circle ($P$). I wish ...
Yoshua Moore's user avatar
1 vote
0 answers
86 views

Suppose $\|u\|=1=\|3v\|$ . If $\|u+v\|=1$ then $u \cdot v=?$

For $\|3v\| = 1$, I isolated $v$ and got: $\|v\| = 1/3$. I used $\|u+v\|^2 = (u+v)\cdot(u+v)$ and got to $$\|u+v\|^2 = \|u\|^2 + 2(u \cdot v) + \|v\|^2$$ $$(1) = (1) + 2(u \cdot v) + (1/9)$$ $$u \cdot ...
Hagar Ali's user avatar
3 votes
0 answers
169 views

Find the indeterminate values of $x_1$ and $y_1$ if $\vec x=(x_1,-2,1,-1)$ and $\vec y=(-2,y_2,-1,-2)$ and $\lVert \vec x\rVert=2\lVert \vec y\lVert$

Let $\vec x=(x_1,-2,1,-1)$ and $\vec y=(-2,y_2,-1,-2)$, wich satisfies $\lVert \vec x\rVert=2\lVert \vec y\lVert$. Find the indeterminate values of $x_1$ and $y_1$. So, assuming $\lVert _\dot{} \lVert ...
Roma_Rayado's user avatar
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1 answer
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Cross product - how do I know this landing direction is the cross product of the other two? [closed]

Here is an image: I am given $\vec{n}$ and $\vec{v_A}$. Apparently, $\vec{v_L}$ is perpendicular to the cross product of $\vec{v_A}$ and $\vec{n}$ but I am struggling to see this even with the ...
PhysicsMathsLove's user avatar
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38 views

Calculate Up Vector of Object on Surface Given Points and Normals

Context I want to find the up vector an object would have if it were leaning on a surface, given a large amount of points on that surface and their associated surface normals (i.e., an equation that ...
natSegOS's user avatar
2 votes
1 answer
43 views

How can I find the closest points that form a perpendicular?

I have three points $A, B, C$ and I would like to find the three closest points $A', B', C'$ that Cause $\vec{A'B'}$ to be perpendicular to $\vec{B'C'}$ and Minimize the sum-square error between ...
Ben's user avatar
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-1 votes
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in a triangle OAB,the point C divides vector AB in the ratio 2:3 and D is the midpoint of vector OB. Find vector OC in terms of a and b.

in a triangle OAB,the point C divides vector AB in the ratio 2:3 and D is the midpoint of vector OB. Find vector OC in terms of a and b using diagrams
Koomson Creative Videos's user avatar
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1 answer
59 views

Difference between rotation matrices [closed]

I am trying to learn about rotation matrices, but when watching tutorials on youtube and even looking at rotation matrix questions on stack exchange I see two different matrices used as general ...
bread.thief's user avatar
2 votes
3 answers
53 views

Help with unit vector problem. [closed]

I am trying to solve this problem: Find two unit vectors that make an angle of 60Β° with v = 4i + 3j. (Round your answers to four decimal places.) I have identified that I need to do systems of ...
baron's user avatar
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0 votes
1 answer
70 views

High School Pentagon Vectorial Geometry question [closed]

I’ve stumbled upon this problem which I thought was simple, but as I was trying to solve it I got stuck. β€œThe convex pentagon ABCDE is considered with the property that AB is parallel to CE, BC is ...
Andrew's user avatar
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3 votes
1 answer
88 views

Interpretation of change in direction cosines of a variable line: Pythagorean theorem for small angles?

Consider a variable rotating line passing through a fixed point. The angle between two successive/adjacent positions of the line is a small angle $\delta \theta$. If the change in the direction ...
Cognoscenti's user avatar
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2 answers
52 views

Vector Identities proof question, using lagrange's formula

How to proof: $$(\vec{A} \times \vec{B}) \cdot (\vec{C} \times \vec{D}) = (\vec{A} \cdot \vec{C})(\vec{B} \cdot \vec{D})-(\vec{A} \cdot \vec{D} )(\vec{B} \cdot \vec{C}) \space\space\space\space (1)$$ ...
Nero's user avatar
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0 answers
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Intuition for vector calculus

In my statistics class, I was introduced to Fisher Information. As it comes from the Taylor Expansion in vector form, I wanted to know terms were ordered in a certain way - whether it was just to make ...
Jackanap3s's user avatar
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0 answers
26 views

How do I get a Unit Vector Representing the Facing of a Quaternion?

I have a point that has a rotation, the quaternion $(.5, .5i, .5j, -.5k)$. If I place this at the origin of a unit sphere, how can I project a vector from the origin to the point on the unit sphere ...
UpTide's user avatar
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1 vote
1 answer
146 views

What are some linear operators outside linear algebra or analysis?

In my study of linear algebra, I learned that the definition of linearity is: A transformation (also: map, mapping, function, etc.) is linear if it preserves additivity and scaling (also: homogeneity)....
user avatar
1 vote
1 answer
93 views

Hello, I need help with this vectors geometry problem

Let $A_1 A_2 \ldots A_n$ be a regular polygon with $n$ sides inscribed in a circle with center $O$. Calculate the sum of vectors $OA_1 + OA_2 + \ldots + OA_n$ when $n$ is odd. I’ve tried this for a ...
Victor Ban's user avatar
-1 votes
0 answers
57 views

Solve equations with vector products

I want to solve the following equations for $x$: $$ x^\top(Ax - b) = 0 \qquad x^\top c = 0 $$ One solution is to set $x = 0$. Are there any other solutions for $x$
shani's user avatar
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1 vote
2 answers
84 views

Does $(x_1v = x_2v) \implies (x_1 = x_2)$ where $x_1, x_2 \in \mathbb{R}$ and $v \in V$?

If $x_1, x_2 \in \mathbb{R}$ are any scalars and $v \in V$ is any vector in a vector space $V$, is it true that $$ (x_1v = x_2v) \implies (x_1 = x_2) $$ I guess this would be termed a cancellation ...
Paul Ash's user avatar
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-1 votes
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Can you find the angle between two planes with the dot product of two vectors in the planes, or do the vectors have to be normal to the planes?

If I am given two planes and told to find the angle between them, why can I not take the dot product of two vectors inside the plane instead of the normal vectors? Why do both not give me the same ...
mvsiri's user avatar
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0 answers
47 views

Autocorrelation of a list of 3D vectors.

I obtained a list of $\overrightarrow{r}_{end-to-end}$ from a Monte Carlo simulation of polymer movement. ...
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