1
vote
0answers
26 views

linear algebra question

Consider $n$ convex polytopes $S_1, \cdots, S_n$ and a set of matrices $W$ such that each matrix $A\in W$, we have that the $i$-th row of $A$ is a member of $S_i$. (In general $W$ is infinite.) ...
0
votes
1answer
33 views

Parallel lines on an ellipsoid

I have an earth modeled as an ellipsoid. I have some line geometries at many places on this model. I have the x,y,z values of all the points on these lines/curve segments. My question is, is there any ...
0
votes
1answer
14 views

How can I prove this vector algebra statement

I am not sure this is true by the way. Given two vectors in D-Dimensional space, A and B is the square of the euclidean distance between them A.A + B.B - 2 A.B ?
0
votes
1answer
23 views

Rotating a plane defined by a normal and a distance from the origin around an arbitrary point in 3D space

I have a plane defined by its normal and its distance from the origin. I have a rotation matrix and a point in 3D space around which to do the rotation. What formula will allow me to do the rotation? ...
0
votes
0answers
29 views

3D Vector projection on a Plane

I want to Project a Vector on to a Plane. Assume, you have a Central Point (1,1,1) and you want to move (0,0,3) in z-direction. How can I project the end of this movement (point) on a plane with ...
0
votes
2answers
25 views

Considering a convex polygon lying on a plane in 3D space, how can I know if a point on that plane lies inside or outside that polygon?

I have a plane in space and a polygon in it. I know the position of each vertices making the polygon. I also know the position of the point on the plane. How can I know whether the point is inside or ...
1
vote
1answer
32 views

A question on the rectangular region defined for a vector in $\mathbb{R}^N$

Let $K = (k_1,k_2,k_3,...k_N)$ be a vector in $\mathbb{R}^N$, consider the region $S_K$ consisting of all vectors $L = (l_1,l_2,l_3,...l_N)$ such that, $|l_i| \le |k_i| \forall i \in \{1,2,3,...N\}$. ...
5
votes
1answer
90 views

what is vector $(\vec{a}\cdot \vec{b})\vec{c} + (\vec{b}\cdot \vec{c})\vec{a} - (\vec{c} \cdot \vec{a})\vec{b}$

Suppose we have three non orthogonal vectors in $R^3$ as $\vec{a}, \vec{b}, \vec{c}$. The vector of $(\vec{b}\cdot \vec{c})\vec{a} - (\vec{c} \cdot \vec{a})\vec{b}$ is in the plane spanned by ...
5
votes
1answer
59 views

What series of 'hyperpolyhedrons' do exist? Is there an effective way to derive their cross-sections by 3-d subspace?

There are two obvious series of 'hyperpolyhedrons'. 'Hyperoctahedron' with vertices $(\pm1,0...0), (0,\pm1,0,...0)...(0,...0,\pm1)$ and each vertex connected by an edge with each other vertex ...
1
vote
0answers
29 views

Cartesian to geodetic conversion of 3D bounding box - How to calculate latitude and longitude from an axis aligned bounding box

I have a geometry with its vertices in cartesian coordinates. These cartesian coordinates are the ECEF(Earth centred earth fixed) coordinates. This geometry is actually present on an ellipsoidal model ...
0
votes
1answer
49 views

Is $\sin \theta_{xy}\leq \sin \theta_{xz}+\sin\theta_{yz}$, where $\theta_{ab}$ is angle between unit vectors $a$ and $b$?

Suppose $x,y,z\in\mathbb{R}^n$ are unit vectors. The angle between unit vectors $a$ and $b$ is $\theta_{ab}=\arccos(a\cdot b)$ where $a\cdot b$ is the dot-product. Is $\sin \theta_{xy}\leq \sin ...
4
votes
1answer
47 views

Project a vector onto the intersection of surfaces

I want to project a vector $\vec v$ onto a surface $S$ defined as the intersection of other surfaces. For example, in 5-dimension I have the surface $S(x_1,x_2,x_3,x_4,x_5)=c$, defined by the ...
0
votes
1answer
22 views

Translation of basis for a vector space on the specified distance

In the Euclidean space $XYZ$ is a basis $X_1Y_1Z_1$ defined that is specified by the vectors $\overrightarrow {O_1X_1}$, $\overrightarrow {O_1Y_1}$ and $\overrightarrow {O_1Z_1}$. How to calculate ...
0
votes
1answer
15 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
14 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
31 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
89 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
36 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
53 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
55 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
11 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
192 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
28 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
28 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
358 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
500 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
35 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
36 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
122 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
186 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
52 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
131 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
52 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
208 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
146 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
39 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
97 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
68 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
98 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
50 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 ...
19
votes
3answers
789 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
54 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
82 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 $\cos(\theta)$ = $\cos A.\cos A'+ \cos B.\cos B'+ \cos C.\cos C'$. ...