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.
12,787 questions
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Determine the reflection line about which two points reflected will lie on two given lines respectively
You're given two lines in the $xy$ plane, let's say
$ Line 1: a_1 x + b_1 y + c_1 = 0 $
and
$ Line 2: a_2 x + b_2 y + c_2 = 0 $
In addition you're given two points $P = (p_1, p_2) $ and $Q = (q_1, q_2)...
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Congruent triangle to a given one, with its vertices at specified distances from the original triangle vertices
You're given $\triangle ABC$ with known vertices in the $xy$ plane. The coordinates of $A,B,C$ are known. Now, given three distances $d_1, d_2, d_3$. You want to determine all congruent triangles $\...
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Vector space of vectors [closed]
I have a question that is really confusing for me. In Serge Lang, the elements of $K^n$ where $K$ is a field are called vectors. The thing is that the vector $V=(v_1,...,v_n)$ belongs indeed to $K^n$ ...
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How to adjust a vector so that both its net circulation and flux vanish?
I am working on a special type of problem which requires that a vector field derived from previously computed fields satisfy 2 constraints:
$$1) \oint \textbf{V} \cdot \hat{\textbf{n}} ds = 0$$
$$2) \...
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59
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From two vectors I can get the cosine of their angle. But how to get that angle value (looking at quadrant) if they have more than two dimensions?
If I have two vectors $\vec{A}$ and $\vec{B}$ having two components I can calculate their scalar product.
And then $\frac{\vec{A}\cdot\vec{B}}{||A|| \times ||B||}$ gives me their cosine,
and through ...
- 269
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76
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When I consider a vector directly, or by its scalar product with $\vec{i}$ I don't receive the same angle measure
I believe that on an orthogonal coordinate system $(O,\vec{i},\vec{j})$, where $(O,\vec{i})$ would design the East, if someone gives me a single vector $\vec{R}(7.6, -3.4)$ it be convenient if I ...
- 269
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Rotating a unit vector to another vector using two consecutive axes
Given a unit vector $u$ and another unit vector $v$, I want to rotate $u$ into $v$ in two stages. In the first stage, I rotate $u$ about a given axis $a_1$ (by an unknown angle) to produce a vector $...
- 188
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3
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Cartesian equation of X axis [closed]
In my school module it is written that the cartesian equation of $x$ axis is
$$
\frac{x}{1}=\frac{y}{0}=\frac{z}{0}
$$
Isn't dividing by zero not allowed? How have they written this equation
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111
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Rotate vector within plane by given angle
Let two 3D unit vectors $V, V'$ be given. Derive vector $W$ created by clockwise rotating $V'$ by angle $\theta'$ around the origin within the plane with normal proportional to $V \times V'$.
I tried ...
- 357
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Vector tangent to meridian
Let there be two different points $ \vec{p_1}, \vec{p_2}$ on a unit sphere.
I need to get unit vector $\vec{t}$ at the point $\vec{p_1}$ tangent to the meridian (big circle) connecting these points.
...
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53
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Simple Algebraic Derivation of the Cross Product [duplicate]
I'm looking for a simple algebraic derivation of the cross product formula: $\vec{a} \times \vec{b} = \|\vec{a}\| \|\vec{b}\| \sin(\theta) \vec{n}$.
I need the derivation to be simple, understandable ...
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Bearing angle of great circular arc between Ottawa Canada, and Sarajevo, Bosnia
I have this problem that I have been working on today. I want to calculate the local direction of the great circle connecting Ottawa, Canada, and Sarajevo, Bosnia. I assume Earth is perfectly ...
user1709783
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Show that $pr_d(\overrightarrow{u}+\overrightarrow{v}) = pr_d\overrightarrow{u} + pr_d\overrightarrow{v}$
On a plane, give a line $d$ and the vectors $\overrightarrow{u}, \overrightarrow{v}\ne\overrightarrow{0}$ such that $\overrightarrow{u},\overrightarrow{v}$ are not perpendicular to the line $d$. Let $...
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I don't believe me, $\overrightarrow{Marc}(1.80\ m, 71\ kg, 56\ y.o.)$ I'm a vector of $\mathbb{R^3}$. But of $\mathbb{R^{+3}}$ or better. Am I right?
My beliefs:
At school, I've seen all the time definition of vectors in $\mathbb{R^n}$.
I've understood that if some are defined in $\mathbb{R^3}$ it means that:
they all have three components
all of ...
- 269
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Why is the $z$ partial derivative of $z = f(x,y)$ equals to $-1$?
We know that a vector $(a,b,c)$ is perpendicular to a plane iff the equation of the plane is equivalent to
$$ax+by+cz+d=0$$
We also hear that "the gradient of $f$ is perpendicular to its tangent ...
- 2,080
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87
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Question on a collinear configuration on quadrilateral
I faced the following problem when fooling around with quadrilaterals.
Let $ABCD$ be a convex quadrilateral. On the edge AB, CD, pick E, F such that $\dfrac{EB}{EA} =\dfrac{FC}{FD}$. Let $M, P, N$ be ...
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$\vec{C} = \vec{A} - \vec{B}$. Scalar product: $\hat{\vec{A},\vec{B}} = 63.4$°, $\hat{\vec{A},\vec{C}}$ = 90° but $\hat{\vec{B},\vec{C}} = 153$°
From a $\vec{A}$ and a $\vec{B}$ vector, I'm trying to extract all the useful values and indications that can be gained through the calculation of their scalar product. And I'm impressed that they are ...
- 269
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Calculate the angle between 3-dimensional lines [closed]
I have two 3-dimensional line segments ($a$ and $b$). The end point of line segment $a$ is the starting point of $b$. Given are their lengths when projected on a horizontal plane ($d_a$ and $d_b$) as ...
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I cannot verify $\vec{A}\cdot\vec{B} = (A_x\vec{i} + A_y\vec{j}+ A_z\vec{k})\cdot(B_x\vec{i} + B_y\vec{j}+ B_z\vec{k})$
According to my book, for a scalar product, this relation is true:
$$\vec{A}\cdot\vec{B} = (A_x\vec{i} + A_y\vec{j}+ A_z\vec{k})\cdot(B_x\vec{i} + B_y\vec{j}+ B_z\vec{k})$$
But I cannot verify it on ...
- 269
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96
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How to find volume of solid body which is formed by right-handed triplet of vectors in first coordinate octant?
I want to find volume of shape on picture below. Red vector is $\vec{a}$, green is $\vec{b}$ and blue is $\vec{c}$. Vectors are right-handed and located in first coordinate octant. I've suggested that ...
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Question on integral over two lines
Question
Thank you for reading my question!
I am trying to solve the problem
$$
\int_T\int_S\frac{1}{|r|^2}dsdt,
$$
where $r(s,t)$ represents the distance between two points $s$ and $t$ on the line ...
- 395
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127
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How to find volume of solid body built on 3-dimensional vector?
How this shape called and why this is not a pyramid? Here link to Desmos3D.
I wanted to find a volume of shape on 3D-vector $ v=(v_1, v_2, v_3) $ and thought it's a pyramid with volume $ \frac{1}{3} |...
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Paradoxical situation arises when I take projection of a vector on its perpendicular.
If two vectors are perpendicular and If we take an arbitrary unit vector in between them and take the projection on the arbitrary vector then use that projection to take projection on the ...
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How to solve this integration regarding cross product and norm?
The problem
Thank you for reading my problem!
Suppose we have two vector functions, the first one is
$$
\mathbf{r}=(a_xr+r_x,a_yr+r_y), r\in(0,1)
$$
The second one is
$$
\mathbf{s}=(b_xs+r_x,b_ys+r_y),...
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Is there a $n$-sided polygon with side lengths $1, \dots, n$ and equal angles, where $n$ is the power of a prime?
This question arose while solving the following problem:
Prove that for every positive integer $n$ such that $n$ isn't the power of a prime, there is a $n$-sided polygon with all its angles equal and ...
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Set Theoretically, NOT CONCEPTUALLY, are Position Vectors the same as Points
TO BE CLEAR: I am asking from a mathematical purist, set-theoretic, construction of math point-of-view, not an applied point of view.
Does $(1,2,3)=\langle1,2,3\rangle$?
If we disect a euclidean ...
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Problem with vector calculus - gradient in spherical coordinates
While deriving the electric field from a dipole source, from the notes I am following I am required to process the following vector operation:
$$
\nabla \left(\frac{e^{jkr}}{r}\mathbf n\cdot \mathbf p\...
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Checking whether a circle of radius a is positively oriented using the fact the interior is on the left when traversing it
Basically, I'm reading Tapp's Differential Geometry of Curves and Surfaces and I'm supposed to show some examples of positively oriented curves to my professor who is the 'advisor' of our sort of ...
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Range of $\cos x$ + $\cos y$ + $\cos z$ where $x,y,z$ are angles between three vector
So, I was solving the following problem
If $\vec{a},\vec{b},\vec{c}$ are unit vectors, then the maximum value of $$|\vec{a}- 2\vec{b}|^2+|\vec{b}- 2\vec{c}|^2+|\vec{c}- 2\vec{a}|^2$$
I simplified it ...
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Linear dependency of 4 vectors in 3D space
Suppose we have 3 vectors $(a, b, c)$ in the x-y plane and a fourth $(d)$ in the y-z plane. If 4 vectors in 3D space are always linearly dependent, how do we express the fourth in terms of the other 3?...
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How do i modify these skewed vectors to intersect?
Essentially, my question is to modify my current equations (that are skewed for now) to set up a situation where one drone intercepts the other, to find k. where one drone is assigned as an “attack ...
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Let $A\equiv (3,5,4)$, $B\equiv (4,3,5)$ and $P\equiv (a,b,0)$. If P be such that $\angle APB\in[0^{\circ},180^{\circ}]$ is maximum, find $a$ and $b$
Let $A\equiv (3,5,4)$, $B\equiv (4,3,5)$ and $P\equiv (a,b,0)$.
If point P be such that $\angle APB\in[0^{\circ},180^{\circ}]$ is maximum,
then find the value of $a$ and $b$.
My Attempt:
If $P$ lies ...
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May I declare a function $f(\vec{u})$, or shall I declare it with a matrix $n \times 1$ parameter, a list of variables, or multiple parameters?
During my school time, I've learned $f(x) = ...$ functions having a single variable for scalar parameter.
Is it possible to declare a function $f(\vec{u}) = ...$ ?
Or isn't it allowed, and I have to ...
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Rotation of a vector in 3D
The cosine of an angle between two vectors X and Y is defined as follows:
\begin{equation}
\cos \angle (X,Y) = \frac{X \cdot Y}{\lVert X \rVert \lVert Y \rVert} (1)
\end{equation}
Let us consider an ...
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Physical interpretation of the curl of a vector field in fluid dynamics and electrodynamics
I have mainly studied the concept of curl in Electrodynamics, like
\begin{equation}
\boldsymbol \nabla \times \mathbf{F} = \left( \frac{\partial F_z}{\partial y} - \frac{\partial F_y}{\partial z} \...
- 8,896
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Finding the equation of the reflection of a line in a plane
As a refresher, I am trying to answer the following question from A-Level Further Mathematics Vectors (I completed my A-Level a few years ago):
Here is my working:
$$\frac{x-2}{4}=\frac{y-4}{-2}=\frac{...
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Constricting a line within a room (3D Plane)
For my assignment, the following scenario has been proposed:
Two bees are placed within a room with skewed paths, with the position vectors given:
$$\text{Bee A =} \begin{cases}
x=200+542.19t\\...
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1
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Explicit calculation of circular symmetry (in relation to electrostatics and inverse square laws)
So to explain what I mean; whenever you have a closed circular boundary, primarily a closed surface or a closed line, and the boundary represents a certain quantity which reduces with the inverse ...
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What is variance in context of projection of a vector?
I looked at the rules, I think it's not wrong to ask people to just explain something to you. (I hope the post won't get flagged). My knowledge level is as much as high-schooler. But I've been ...
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Accepted name and tensor notation for a form
Suppose there is a function $ f $, mapping a vector space over $ F $ into itself $ f: F^n \rightarrow F^n $, defined like this:
$$ f \left( \mathbf{x} \right)_i = \sum\limits_{j=1}^n \sum\limits_{k=0}^...
- 123
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Show that $A_nA_{4n} \leq 3 \sqrt{n},$
The problem
Let O be an arbitrary point and the sequence $(A_n)_{n\geq 1}$ of points in the plane, such that $OA_1=1, A_nA_{n+1} \perp OA_n, A_nA_{n+1}=OA_1=1$ for any $n \in N$. Prove that:
a) $A_nA_{...
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3
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117
views
show that lines MA',NB',PC' are concurrent
the problem
Let $O$ be a point in the plane of triangle $ABC$ and $M$, $N$, $P$ the midpoints of sides $BC$, $CA$, $AB$. If points $A'$, $B'$, $C'$ lie on segments $AO$, $BO$, $CO$, such that $\...
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determine the measure of the angle $\angle {ABC}$
the problem
Triangle ABC is inscribed in the circle with diameter BC. The tangent at A to the circle intersects the tangents at B and C to the circle at points D and E, respectively. Knowing that $CD \...
- 1,199
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1
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Is this subset of $\Bbb R^3$ really a subspace? ChatGPT seems to contradict MIT lecture. [closed]
I'm reviewing the definition of subspaces in linear algebra, and I'm having some confusion identifying whether a certain subset of $\mathbb{R}^3$ qualifies as a subspace.
Here is a set defined in ...
- 25
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1
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87
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Transposition of Matrices
My question concerns understanding the transpose as an operator acting on the dual of a space rather than on the space itself.
This is a paraphrased version of Chapter 4 in Linear Algebra by Lax.
Let $...
2
votes
1
answer
137
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Dot Product Geometric Form
$$
a\cdot b = |a|\cos(\theta)|b|
$$
The dot product is often defined as the "measure of how similar two vectors are".
For that isn't the sole term $|a|\cos(\theta)$ enough? I don't ...
- 21
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votes
1
answer
65
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New Conjecture Found In Geometry. Related to tetrahedron and vector 3D. Please review the statement and provide views. [duplicate]
Hi guys currently working on vectors I studied incenter of tetrahedron and the inradius for a tetrahedron was (3*volume)/surface area in modulus.
I checked for some other closed figures and the ...
0
votes
2
answers
53
views
Can you check if a collection of eigenvectors are unique to a matrix of some form? [closed]
If you have a collection of n (nonzero and unique) eigenvectors, is there a way to find a general form of an n-by-n matrix that corresponds to them? In addition, is it possible to check for some kind ...
- 117
1
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1
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66
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Equation of a line of reflection of two points in another point
Three points with known coordinates are given - $P_1 = (x_1,y_1)$, $P_2 = (x_2,y_2)$ and $P_3=(x_3,y_3)$. The line $y_l(x)$ goes through a point $P_2$. A point $P_a$ is an intersection point of lines $...
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0
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130
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Is there a name for the operation like the dot product, but with the add/multiply order swapped
The dot product, as we all know, is this:
$$
\mathbf{a} \cdot \mathbf{b} = \sum_{i=0} \mathbf{a}_i \mathbf{b}_i
$$
But I've recently been dealing with an operation like this, where the sum and product ...
- 273