-1
votes
0answers
35 views

Evaluating the distance beween two vectors of different dimensions.

I am asking for a way (if possible) to evaluate the distance between two vectors $p=(p_1,....,p_{n})∈ℤ^{n}$ and $q=(q_1,....,q_{m})∈ℤ^{m}$, where $n \neq m$.
0
votes
1answer
14 views

Eigenvector shared by two endomorhisms

I am guessing if the following fact is true: Let be $V$ a finite vector space above a field $K$. Let $f, g$ be two endomorphisms of $V$ with $f g = g f$. We assume that both $f$ and $g$ have got at ...
0
votes
1answer
12 views

Lines perpendicular to vectors- are they similar triangles?

Please excuse my horrible vector drawing skills. Let us first assume that we have a third vector, called $\Delta V = V_2 - V_1$ Now, these three vectors make a triangle, $V_1, V_2, \Delta V$. Let ...
0
votes
2answers
23 views

Finding a vertex to complete a parallelogram in $\mathbb{R}^3$ and finding a cosine between two vectors.

Trying to solve an exercise regarding vectorial geometry, I have two doubts: For $A,B,C,D \in \mathbb{R}^3$, $$A = (0,1,0)\\ B = (2,2,0) \\ C = (0,0,2) \\ D = (a,b,c)$$ First, determine ...
0
votes
1answer
37 views

Dimension of Hyperplane

Why the dimension of a of N dimensional space hyperplane is N-1? Is there a mathematical ...
1
vote
4answers
88 views

How to prove U•V = |U|•|V|cos(θ), if θ is the angle between |U| and |V|

This is a snippet from my book. How did they get from |U|$^2$ = U • V = |U|•|V| |U|/|V| ?
0
votes
0answers
17 views

Grab major changes along a line

I have a line where there are many points along it. I want to be able to find the major changes in the line and just grab those points rather than grabbing all the points the line has. Here's a quick ...
1
vote
2answers
29 views

Writing a vector as the sum of two other vectors.

Suppose you have 2 vectors $\vec a = (1,1,2)$ and $\vec b = (3,4,-2)$, how would you write $\vec a$ as the sum of 2 vectors $\vec c$ and $\vec d$ where $\vec c$ is in the direction of $\vec b$ and ...
1
vote
1answer
47 views

Show that a square with vertices t, u, v, w has center 1/4 (t+u+v+w).

I need a help with this question! Show that a square with vertices t, u, v, w has center 1/4 (t+u+v+w).
0
votes
1answer
51 views

Curl and unit vectors after a change of coordinates

Let $x,y,z$ be a system of coordinates with unit vectors respectively $\mathbf{u}_x$, $\mathbf{u}_y$, $\mathbf{u}_z$. Moreover, we have $\nabla = \displaystyle \frac{\partial}{\partial x} ...
0
votes
1answer
10 views

Symmetric Parallelograms Under Linear Transfer Marticies

I am trying to show that a parallelogram which is symmetric about the origin stays symmetric about the origin under the action of a linear transfer matrix. It is a fairly trivial case to draw a ...
4
votes
1answer
183 views

Axiomatization of angle measuring in real vector spaces

In linear algebra / analytic geometry it is common to define the angle between two vectors $u,v \in V$ of an euclidean vector space $V$ by $\angle (u,v) := \arccos \frac{(u,v)}{\|u\| \cdot \|v\|}$. Is ...
1
vote
0answers
27 views

How to calculate normal (of magnitude 1) of a triangle?

I am currently doing a bit of geometry practice and wanted to know how to calculate the normal (of magnitude 1) of a triangle defined by 3 vertices: a, b and c`. ...
1
vote
1answer
33 views

Dual Basis problem

I've been dealing with this but I haven't been able to understand the underlying principles of dual basis, so i don't know how to do it well. It starts like this: Have $(e_1, e_2, e_3)$ basis of the ...
0
votes
1answer
27 views

Sequence of sets and vectors

Let $A(0,1),B(0,0),C(1,0)$ and $D(1,1)$ be four points in the plane $xOy$. Define $M_3=\{A,B,C\}$ and $M_{n+1}=M_n \cup \left\{Z\epsilon xOy\mid \exists V,W\epsilon M_n\text{ for which ...
-1
votes
1answer
114 views

How to use geometry to express unit vectors of spherical coordinate system in terms of Cartesian unit vectors

It's quite easy to express unit vector $\hat{r}$ in sum linear combinations of Cartesian unit vectors $\hat{x}$, $\hat{y}$ and $\hat{z}$. But I am not sure how I can use geomtery to find a Cartesian ...
0
votes
2answers
235 views

Do four dimensional vectors have a cross product property? [duplicate]

We know how to make cross product of three dimensional vectors. $$ \vec A \times \vec B = \vec C$$ Where : $ \vec A = (A_i; A_j; A_k)$ $ \vec B = (B_i; B_j; B_k)$ $ \vec C = (C_i; C_j; C_k)$ $C_i = ...
0
votes
1answer
34 views

Problem with plane and angles

I have the non coplanar straight lines that touch in (1, -2, 3): $$L1: \frac{x - 1}{2} = \frac{y + 2}{2} = \frac{z - 3}{1}$$ $$L2: \frac{x - 1}{3} = \frac{3 - z}{-4}; y = -2$$ $$L3: \frac{x - 1}{2} ...
0
votes
1answer
32 views

Straight lines forming an equilateral triangle

I have the straight lines: $$L1: (1, 0, 0) + r(1, 1, 1)$$ $$L2: (7, 4, 3) + s(3, 4, 2)$$ I'm asked to get the vertices of the equilateral triangle of side 2 * 2 ^ (1/2) so one vertex belongs to L2 ...
0
votes
3answers
47 views

Determining if a point is inside two planes

I have two planes(Plane 1 and Plane 2) the normals ($n_1$ and $n_2$) of which are known to me. How do I determine if a point is inside the two planes? By inside I mean the 3d space between Planes 1 ...
1
vote
0answers
32 views

Have another question about straight lines

I made a question about this topic some hours ago and i found another problem that i can't solve. I hope this is not against the rules of "homework". So i have the following lines: $$L1: P1(1, 1, 2) ...
1
vote
1answer
34 views

I have some problems with straight lines and planes

Firstly, I need to say that English is not my first language and the problems were written in Spanish. I have never read a Math problem in English, so some words may be confusing. If they are, please ...
2
votes
2answers
99 views

check if point is on a plane (using Heron formula ?)

Is this true that if any of parameters a, b, c, d is equal to sum of three others then 4 points are on same plane? I am given 4 points in 3 dimensional space. Is this correct to state that all 4 ...
0
votes
2answers
123 views

Find tangent vector to surface given a point on the surface and its normal vector (for a sphere)

I need to know how to find a tangent vector to a point on the surface of a sphere if I am given the point P and the normal vector at that point N. I know that there are many possible tangent vectors ...
0
votes
2answers
51 views

Is every tensor an element of a vector space?

As, the tensor product of two vector spaces $V$ and $W$ over a field $K$ is another vector space over $K$, is it true to say that every tensor is an element of a vector space ? (if we do not consider ...
3
votes
2answers
114 views

What is (fundamentally) a coordinate system ?

Consider the following construction of vectors and points. Let's start with a vector space, or more specifically a coordinate space $F^N$ over a field $F$ and of $N$ dimensions. The elements of this ...
1
vote
0answers
46 views

Computationnal geometry: vector, basis, point and coordinate system?

I am trying to build a small geometrical library in C++, that is mathematically consistent (not so false). The goal here is to construct two concepts: vectors and points. I am not sure that the ...
1
vote
1answer
145 views

Rigorous definition and relations between point/vector/affine space/vector space/basis/frame/coordinate system

I am trying to understand the exact relation between all these things: point vector affine space vector space basis frame coordinate system Can you explain me rigorously (in the mathematical ...
1
vote
0answers
141 views

Computing Euler Angles from Direction Cosines Vector

My problem simply as the following: Suppose we measured the orientation of a plane object with respect to a reference fame. (where the reference frame parallel to plane frame). The unit normal vector ...
0
votes
1answer
38 views

Missing component of a 4D vector

I need to calculate the missing component of a 4D vector, when I know that one of the dimensions is always positive and less than or equal to the magnitude. In other words, I have four variables x, ...
0
votes
1answer
77 views

4D-vector calculations

For a 4D vector, how can I calculate any component as a function of the three other components and a magnitude and vice versa? I want x = f(y, z, i, m) where m is the magnitude of the vector. Will ...
2
votes
2answers
66 views

Equation of plane without cross product

We know that vectors $(3,3,4)$ and $(-1,-1,5)$ span a plane in $\mathbb{R}^3$. Can we somehow readily infer that the plane's equation is $x_1 - x_2 = 0$? Cross-products have not yet been introduced ...
0
votes
1answer
92 views

Converting from spherical coordinates to cartesian around arbitrary vector $N$

So if I'm given an arbitrary unit vector $N$ and another vector $V$ defined in spherical coordinates $\theta$ (polar angle between $N$ and $V$) and $\phi$ (azimuthal angle) and $r = 1$. How do I ...
1
vote
0answers
32 views

Geometric accuracy analysis of 2d rectangular models

I have reconstructed set of rectangular objects lie on a 2D plane (for ex. ABCD). All these objects are in a one coordinate system. On the other hand, I have reference models for all of them ...
0
votes
0answers
48 views

Number of ways to cut a square

How many ways are there to cut the unit square into two pieces? And how many ways are there if the two pieces must have equal area? Some special cases: A. If the cut is required to be a horizontal ...
18
votes
3answers
719 views

Determinant of transpose?

$$\det(A^T) = \det(A)$$ Using the geometric definition of the determinant as the area spanned by the columns could someone give a geometric interpretation of the property? Thanks!
1
vote
1answer
50 views

Coordinates relative to arbitrary 3D plane

Say that I have an arbitrary plane, $\mathcal{P}$, in $\mathbb{R}^3$ that is defined by a given vector, $\vec{v}_0$, on the plane and a normal vector, $\vec{n}$. I will be using $\vec{v}_{0}$ as a ...
2
votes
2answers
81 views

Angle between vectors of the form $(\cos A,\cos B,\cos C)$

The question: Two vectors $S=(\cos A,\cos B,\cos C)$, $S'=(\cos A',\cos B',\cos C')$, What is the angle between them? The answer is $\cos(\theta)$ = $\cos A.\cos A'+ \cos B.\cos B'+ \cos C.\cos C'$. ...
1
vote
0answers
29 views

What is first edge position in the Minkowski sum of two convex polygons in the plane?

I am trying to understand the informal algorithm of the Minkowski sum of two convex polygons in the plane as described here: ...
2
votes
3answers
78 views

Simple way to parameterize two perpendicular vectors

Given are two vectors in $\mathbb{R}^3$, $\bar{u}$ and $\bar{v}$, such that they are perpendicular ($\bar{u}\cdot\bar{v}=0$) and of equal length ($|\bar{u}|=|\bar{v}|$). Is there a "nice" way to ...
5
votes
0answers
42 views

Generating a 3d ribbon from a series of points

I am attempting to generate a 3d ribbon from a set of 3d points. The idea is to generate a realistic ribbon which follows those points. In its current state, one example looks like this: In this ...
2
votes
1answer
42 views

projective geometry and relationship of cross-ratios

Define for pairwise different points $P_i=[v_i]$ the cross-ratio $\operatorname{CR}(P_1,P_2,P_3,P_4) = \frac{\det(v_1,v_2)}{\det(v_2,v_3)}\cdot\frac{\det(v_3,v_4)}{\det(v_4,v_1)}$ on $\mathbb{KP^1}$ ...
0
votes
1answer
68 views

bijective maps leaving cross-ratio invariant are just the projective transformations

Show the bijective correspondence between a. bijective maps $f: \mathbb{P(K^1)} \to \mathbb{P(K^1)}$ which keep the cross-ratio invariant b. projective transformationes, i.e. $f: \mathbb{P(K^1)} \to ...
1
vote
1answer
112 views

Tensor product, wedge product, Hodge product, dyad, or what?

Suppose I have two vectors $\mathbf{u} = (u_1, u_2, u_3)$ and $\mathbf{v} = (v_1, v_2, v_3)$ in $\mathbb{R}^3$. I can regard $\mathbf{u}$ as a $3 \times 1$ matrix, and $\mathbf{v}$ as a $1 \times 3$ ...
2
votes
2answers
33 views

map colinear triple of points to another triple of points in $\mathbb{R}^2$

Given two triples of pw different colinear points in $\mathbb{R}^2$ so $(x_1,x_2,x_3),(y_1,y_2,y_3) \in (\mathbb{R}^2)^3$. There is a map of the form $T:\mathbb{R}^2\to\mathbb{R}^2,x\mapsto Ax+b$, ...
0
votes
1answer
56 views

Norm, Euclidean Space and Distance

To a complete layman, how would you define the following terms intuitively? $norm$ , $euclidean$ $space$ , and $euclidean$ $distance$ ? Note: I have tagged Linear Algebra and Probability Theory ...
1
vote
0answers
28 views

sweeping edges till they get a given elevation on an oblique plane

I am constructing wireframe model of 3d objects (prisms,..etc.). from a triangular mesh, I have obtained boundary points and fit striaght lines in order to get polygon edges refering to prism ...
1
vote
1answer
33 views

The idea of the transverse to a vector field

I have a quick question. I am independently reading a book on three dimensional geometry and topology. One line has been stumping me. Here is the following paragraph I do not understand: "Let $X$ be a ...
0
votes
0answers
19 views

isolated zero z of X on a star shaped polygon

Consider a polygon that is star shaped with respect to the isolated zero $z$ of $X$. I want to show that the boundary of the polygon can be made transverse to $X$ by jiggling vertices only in the ...
0
votes
1answer
83 views

Correcting plane parameters with the fixed azimuth angles

I am trying to reconstruct specific 3d objects such as cubes, pyramids and so on. For this, i am using point cloud data and then fitting planar surfaces for the segmented point patches. Planes ...