Tagged Questions
2
votes
1answer
28 views
How to Find the Center of a Parallelogram
I want to find the center of a parallelogram in order to use it in my java program. I have four coordinates of the parallelogram and I want to find the center coordinate of the parallelogram. It seems ...
4
votes
1answer
41 views
Book on quadric surfaces with linear algebra
Most information that I can find about quadric surfaces is written from a calculus perspective - without using any matrices or vectors. However, I would like to have a reference that tells me the ...
2
votes
3answers
122 views
On integral of a function over a simplex
Help w/the following general calculation and references would be appreciated.
Let $ABC$ be a triangle in the plane.
Then for any linear function of two variables $u$.
$$
\int_{\triangle}|\nabla ...
0
votes
1answer
22 views
Collinearity in n dimensions
What is the best way to check if $m$ points are collinear in $n$ dimensions?
I mean I have
$p_1=(3, 4, 5, 2),\quad p_2=(6, 3, 4, 2),\quad p_3=(5, 3, 5, 6),\quad p_4=(4, 2, 7, 4)$
or
...
2
votes
1answer
24 views
Equation of third side of Triangle
A Triangle is formed by Pair of lines
$$ ax^2+2hxy+by^2=0$$ and a third side L3.Given the Orthocentre of Triangle is $$(c,d)$$, Find Equation of Third Side.
1
vote
2answers
34 views
How do I find the point between two point at a specific distance (in x,y coordinates)
I am working on a programming project and have run into a problem.
I need to find a point that is on a line. For example if my line is AC and I have the coordinates for point A and C, how can I get ...
-1
votes
1answer
66 views
Vector valued functions [closed]
What is a vector valued function ? Vectors are just $n-tuples$ , and then how are we able to describe geometrical shapes using them ?
Is a vector function such that it outputs $n-tuples$ using other ...
1
vote
0answers
21 views
Convex cone as sum of simplices?
In 3D a pyramid with a square base can be decomposed into the sum of two tetrahedra, i.e. two 3-simplices.
I am dealing with a homogeneous N-dimensional system of inequalities and my solution is a ...
1
vote
1answer
26 views
How to sketch the graph of $\varphi(X)=d(X,A)+d(X,B)$ when $A$ and $B$ are not given?
$A,B$ are points in an axis, disposed in this order. Sketch the graph
of the following function:
$$\varphi(X)=d(X,A)+d(X,B)$$
$d(A,B)$ is the distance from point $A$ to point $B$.
I'm ...
1
vote
1answer
20 views
Question about the dimension of the intersection of two subspaces of a vector space $V$.
Let $M, N$ be two subspaces of a vector space $V$ with dimension $k$. Suppose that $\dim M=m$, $\dim N =n$. It is said that $\dim M \cap N \geq m+n-k$. Suppose that $M, N$ are two parallel planes in ...
-1
votes
0answers
25 views
Imaginary line passing through non-collinear points in R3.
I have come to a problem where n points are provided in 3-Dimensional plane. I need a imaginary line which can be assumed that it is passing through these points.
1
vote
2answers
18 views
What's the geometric interpretation of a semidenifite matrix smaller than identity matrix?
What's the geometric interpretation of a semidenifite matrix in terms of eigenvalues/eigenvectors with the condition:
$$
0 \preceq W \preceq I
$$
0
votes
1answer
46 views
A question about definition of hyperplane
In a course I took, the instructor gave a definition of hyperplane as follows:
Let $X$ be a vector space and $f:X\to\mathbb{R}$ a linear function. Then $M_a=\{x\in X|f(x)=a\}$ is called a hyperplane. ...
4
votes
0answers
48 views
Invariant of matrix under elementary transformations
$\DeclareMathOperator{\rank}{rank}$
Let $A \in \mathbb R^{n \times n}$, $b \in \mathbb R^n$, $c \in \mathbb R$. Consider the following matrix
$$
B = \begin{bmatrix} A & b \\ b^T & c ...
4
votes
1answer
42 views
Optimal rotation to align a circle with external points
I have a circle $C$ with radius $r$ and a set of finite points $P=\left \{ p_1,p_2,\ldots,p_n \right \}$ are identified external to the circle $C$. These points may lie on the exterior or the interior ...
4
votes
1answer
63 views
Angular alignment of points on two concentric circles
I have two concentric circles $C_1$ and $C_2$ with radii $r_1,r_2$ such that $r_1< r_2$and a set of finite points $P=\left \{ p_1,p_2...p_n \right \}$ and $Q=\left \{ q_1,q_2...q_n \right \}$ are ...
0
votes
1answer
26 views
Find Line with specific Angle to another Line
Given any line in 3 dimensional space
$$A: \vec{X} = \vec{O} + \lambda \vec{D}$$
and any angle $\phi$, I want to find another line $B$ which fullfills the following criteria:
it ...
4
votes
1answer
96 views
Name for this simple inequality
Let $x,y$ vectors in $\mathbb{R}^3$. From
$$\Vert x+y\Vert^2\geq 0$$ it follows that
$$2x\cdot y\geq -\Vert x\Vert^2-\Vert y\Vert^2$$
Has this inequality a name?
1
vote
0answers
36 views
Parallelogram law, dot product [duplicate]
Prove that if $||\cdot||$ satisfies $||u-v||^2 + ||u+v||^2 = 2(||u||^2 + ||v||^2)$ , then $u \cdot v = \frac{1}{2} (||u+v||^2 - ||u||^2 - ||v||^2)$ is dot product and $||u||^2 = u \cdot u$.
I've ...
0
votes
0answers
28 views
Expressing vectors in a basis (Octagon)
Consider a regular octagon with the corners (points) $A,B,C,D,E,F,G$ and $H$ (counterclockwise). Let the vectors $\mathbf{AC}$ and $\mathbf{AD}$ be a basis for the plane. Express $\mathbf{AB}$, ...
0
votes
1answer
37 views
Optimizing the area of a triangle in space.
A triangle has two corners, $(8,0,3)$ and $(0,8,3)$ and a third curve in space that consists of all points $(8,8,a^{2}+3)$, where $a$ is a real number. Calculate the area of the triangle as a function ...
1
vote
1answer
25 views
Decomposition of a single 4D rotation
I have a $4\times 4$ matrix $M$ which represents a general 4-dimensional rotation.
$$
M = \pmatrix{a_{11} &a_{12} &a_{13} &a_{14}\\a_{21} &a_{22} &a_{23} &a_{24}\\a_{31} ...
13
votes
1answer
121 views
What is duality?
I have seen some examples of duality. Sometimes applied to theorems, as for example Desargues theorem and Pappus theorem. Sometimes applied to spaces, for example the dual space of a vector space. ...
3
votes
1answer
27 views
Find $ \vec{z} \parallel \vec{x}$ and $\vec{y}-\vec{z} \perp \vec{x}$
I'm trying to solve the following question:
For $\vec{x},\vec{y} \in \mathbb{R}^2$ and $\vec{x},\vec{y} \neq \vec{0}$ find $\vec{z} \in \mathbb{R}^2$ with $\vec{z}\parallel\vec{x}$ and ...
1
vote
2answers
33 views
Why does a weighted sum of complex numbers stay in the convex hull
Given any set of complex numbers (or 2D vectors) $x_1,x_2,...x_n$ and real numbers $w_1,w_2,...,w_n$ if we have $\bar{x} = \frac{w_1x_1 + w_2x_2 ... + w_nx_n}{w_1 + w_2 ... + w_n}$, we will find that ...
1
vote
1answer
41 views
Overdetermined linear system
I am studying the problem of the solution of the following equation: a*P+(1-a)*R=b*Q+(1-b)*T, wher P,R,Q and T are given points in the space. My unknows are a and b, so I have an overdetermined system ...
3
votes
2answers
82 views
Finding a transformation matrix
Please can you help me with the following problem:
I have a rectangle with center $(x_1,y_1)$ and sides $a,b$ where side $a$ is parallel to axis $Ox$. I want to find a transformation matrix that:
a) ...
0
votes
1answer
32 views
Normalized/Unit vector
I am confused with the following -
Do unit vectors and normalized vectors indicate the same thing?
If I have a vector, and if I divide it by the length of the vector, then what will we get?
0
votes
3answers
107 views
Rotation matrix in 3-dimensional space with two angles.
I am trying to find a description of a rotation in a three-dimensional space with a matrix that uses only 2 angles. It is easy to find one which uses three angles, since I can always consider the ...
0
votes
3answers
97 views
Geometrical Proof of a Rotation
I wanna prove geometrically ( and not by linear algebra, doing transformations in the bases ) the result of the rotation of a point. The proof should only include geometrical steps like using ...
0
votes
1answer
26 views
Is the volume of an k-parallelogram determined by its edge lengths and angles
It is easy to show that the volume of a $3$-parallelogram $P$ is purely a function of the edge lengths and the angles, simply by computing the square root of the determinant of the gram matrix of $A$ ...
2
votes
3answers
63 views
Densities of planes
I have misunderstanding about what does mean density of planes. For example I was trying to figure out definition of co-vector, and while browsing in internet definition of it, I found ...
1
vote
1answer
88 views
Find the vector parametric equation of the line through $A$ and $B$.
Let $A = (1; 2; 1)$ and $B = (2;-1;-1)$.
Find the vector parametric equation of the line through $A$ and $B$.
If $C = (3;-4;-3)$, show using your equation that $A, B$ and $C$ are collinear.
I missed ...
1
vote
1answer
58 views
Intersect a line with a bicubic Bezier Surface Patch.
This question mentions Bezier surfaces, but doesn't go into any detail. How do you going about finding the intersection between a line, $E_{pos} + E_{dir}*t$ and a Bezier surface patch, $P = $
...
1
vote
3answers
72 views
Show that $|AB|\textbf{p}+|BC|\textbf{q}+|CA|\textbf{r}=\textbf{0}$ (without using the cross product).
Let $ABC$ be a triangle, and p, q, r be the unit vectors perpendicular to
$AB$, $BC$, $CA$ directed inside the triangle. Show that $|AB|\textbf{p}+|BC|\textbf{q}+|CA|\textbf{r}=\textbf{0}$.
I tried ...
5
votes
0answers
52 views
Volume of the intersection of ellipsoids
How do I compute the volume of the intersection of two n-dimensional ellipsoids?
Given an $n$-vector $c$ and a symmetric positive-definite $n\times n$ matrix $A$, define the ellipsoid ...
1
vote
4answers
96 views
Distance between planes
Find the distance between the planes $$x + 2y +2z = 4$$ $$z= -\frac12 (x-1)-(y-2)+3$$
First of all how do you check if they are parallel?
The integers in plane two are leading me astray? How do I ...
0
votes
1answer
36 views
Definition: product $\alpha \cdot f $ with $ f \in \operatorname{Hom}_K(E,F)$ , $\alpha \in K$
I need the definition of product $ \alpha \cdot f $ with $ f \in \operatorname{Hom}_K(E,F), \alpha \in K $.
Thanks in advance.
P.S. In this case $E$ and $ F $ are vector spaces on $K$.
0
votes
1answer
21 views
Definition: sum $f+g $ with $ f,g \in Hom_K(E,F) $
I need the definition of sum $ f+g $ with $ f,g \in Hom_K(E,F) $...
Thanks in advance
1
vote
2answers
49 views
Find the line passing thought the point $p=(1,2,0)$, paralel to the plane…
Find the line passing thought the point $p=(1,2,0)$, paralel to the plane $P=\{x,y,z \mid x+2y-z=-4\}$ and crossing the line $L=\{(x,y,z):x+2y=2, y+z=4\}$
So I've tried to put the equation of plane ...
0
votes
1answer
50 views
How to use Pythagoras theorem with alternative axes
please see the following picture and tell me if you know the solution. Thank you very much.!
4
votes
0answers
90 views
Geometric intuition for Jordan normal forms (invariant subspaces, shearing, scaling, etc.)
I'm trying to visualize what a linear operator does to a vector space if that operator can be put into Jordan normal form.
For concrete motivation, let's take $V = \mathbb{R}^3$, with some linear ...
1
vote
1answer
49 views
Relation between Hadamard product and scalar product
Is there a known relation/formula for
$$(A\circ B, C)$$
where $\circ$ is the Hadamard product and $(\cdot, \cdot)$ is the scalar (euclidean) product?
In particular, I have a vector $y$ and a two ...
2
votes
2answers
49 views
Duplicate quadratic Bézier curve with new start point?
I have Bézier curve as shown by the wikipedia gif here:
I would like to create a new curve that is a segment of the old one. For example, in this gif (from the same article):
.. if I wanted B to ...
3
votes
2answers
139 views
Computing the decay factor for a full rank wide matrix, or finding a unit vector farthest away from a set of spanning unit vectors
Let $A$ be a tall matrix that is not rank-deficient and has normalized columns. That is $A$ is $m\times n$, $m<n$ and rank$(A)=m$, and $||a_i||_2=1$ for all columns $a_i$.
Define ...
1
vote
1answer
35 views
Limiting Degrees of Freedom in 3D Point Registration
I'm search for some assistance in my application of Arun's algorithm for registration (fitting) of two 3D point sets using the Singular Value Decomposition: ...
0
votes
3answers
86 views
How to translate a point by $90$ degree counter clockwise direction?
I would like to find a point which is rotated $90$ degree counter clockwise direction about the origin. For example, the point $(2,0)$ is taken to $(0,2)$.
Given $(x,y)$, how do i find the new ...
1
vote
2answers
59 views
Equation of plane — point/vector pedagogy
Suppose we have a point $\mathbf P$ and a vector $\mathbf n$ in plain ordinary 3D space. Here I am deliberately using upper-case letters for points, and lower-case points for vectors, since they are ...
0
votes
1answer
26 views
Distance between to points on opposite sides of a plane is greater then distance between the points and the plane
Given two points $P$ and $Q$ on opposite sites of a plane $H: n^T \cdot x = a$. I want to show that $||P-Q|| \ge d(P, H)$.
I know that $d(P, H) = \frac{n\cdot \overline{OP}}{||n||} + a$ and that $n^T ...
0
votes
3answers
72 views
How to calculate triangle-line collision in 3D?
If there is a given triangle (tx1, ty1, tz1), (tx2, ty2, tz2), (tx3, ty3, tz3) and two given point of a line ...
