0
votes
2answers
62 views

Difference between Euclidean space and vector space?

I often hear them used interchangeably ... they are very complicated to make any use of. Wikipedia words: Euclidean space: One way to think of the Euclidean plane is as a set of points ...
1
vote
1answer
49 views

Proving a strange vector inequality in the euclidean space

It seems to hold the following inequality in an euclidean reference frame $(x,y,z)$: $$\overrightarrow{U}\cdot\overrightarrow{U}\ge\sqrt{2}\left(\Omega_x+\Omega_y\right)$$ where: ...
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| ?
1
vote
1answer
49 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
22 views

Find vector resultant in rhombus

Uhm I can't find a solution for this problem, perhaps someone can help me with a hint or a solution, thanks in advance :) $$DG=GH=HI=IG\\and\\ AE=EF=FB$$ Find resultant for U+V+W
0
votes
2answers
66 views

Is there a Taylor series for vector cross product?

I have this equation, where $u,v,w,a,b,Ɵ$ are constants. The RHS comes from the Geometric definition of the LHS $(u,v,w)(a,b,c)=||(u,v,w)||||(a,b,c)||\cos(\theta)$ Expanding the 2-norms ...
0
votes
1answer
71 views

projection onto vector spaces

How do you project a vector on to the euclidean ball? For example, if there is a vector $x ∈ R^n$ how does one project this onto the euclidean ball. What are the steps for projecting a vector onto a ...
0
votes
1answer
27 views

What direction does a vector with more than two entries point at?

Say you are given theses two vectors: u = (1, -2, 4) v = (-2, 4, 8) Since there are three entries, how do you know if they point in the opposite/same/different direction?
1
vote
1answer
170 views

Line-preserving transformations

Is there a name for the class of transformations on the Euclidean plane (or projective plane) that preserves lines? They are not all affine transformations; consider a perspective projection $p$ in ...
-2
votes
1answer
56 views

Are all metric space as a euclidean space?

I believe that all euclidean space is a metric space? But I need to know about inverse? I mean: are all metric space as a euclidean space? Is there any kind of metric space which is not euclidean ...
0
votes
1answer
2k views

Equation of a line passing through a given point, perpendicular with a vector

Find the line that goes through A(1,0,2) and is perpendicular to r = (-2,3,4) + s (1,1,2) I did a bunch of work, but I don't know if any of it is right. I erased most of it, but this is what I came ...
3
votes
1answer
266 views

Finding intersection of 2 planes without cartesian equations?

The planes $\pi_1$ and $\pi_2$ have vector equations: $$\pi_1: r=\lambda_1(i+j-k)+\mu_1(2i-j+k)$$ $$\pi_2: r=\lambda_2(i+2j+k)+\mu_2(3i+j-k)$$ $i.$ The line $l$ passes through the point with ...
0
votes
1answer
43 views

How to fit an object of constant size based on measurements to known points

I'm looking for a mathematical solution for solving where the base of a camera crane (ie a constant square or rectangle of known dimensions) is with measurements to known points. This seems to be a ...
0
votes
1answer
110 views

Questions related to vectors and linear algebra

1) Given vectors $u = (1, 1, -2)$ and $v = (0, 1, 1)$, find the value of $t$ such that magnitude of $u + t(v)$ has the smallest value I have no idea how to begin. 2) Given vectors $u = (1, 1, -2)$, ...
0
votes
1answer
55 views

Norm to evaluate precision

I have two vectors x and y with their respective n coordinates/components and this norm between them: $$ norm = \sqrt{\frac{\displaystyle\sum_{i=1}^{n} (x_{i}-y_{i})^2}{\displaystyle\sum_{i=1}^{n} ...
0
votes
3answers
2k views

Proof of test of collinearity and coplanarity

Statement : If there are 3 points with position vectors a, b and c. Then the points are collinear if and only if there exist scalars x,y,z, not all zero,such that x a + y b +z c = 0 where x+y+z =0. ...
3
votes
1answer
589 views

How to plot N points on the surface of a D-dimensional sphere roughly equidistant apart?

Let's say I have a D-dimensional sphere with a radius R. I want to plot N number of points evenly distributed (equidistant apart from each other) on the surface of the sphere. It doesn't matter where ...
1
vote
2answers
1k views

Solid angle between vectors in n-dimensional space

There is a formula of to calculate the angle between two normalized vectors: $$\alpha=\arccos \frac {\vec{a} \cdot\ \vec{b}} {|\vec {a}||\vec {b}|}.$$ The formula of 3D solid angle between three ...
1
vote
2answers
73 views

Simple Vector Question in $\mathbb{R}^3$

Two points $A$ and $B$ in $\mathbb{R}^3$ with origin $O$ are given in terms of a Cartesian coordinate system by $A = (1, 2, 3)$ and $B = (4, 5, −1)$. How do you find the point $C$, such that $OACB$ ...
0
votes
2answers
57 views

Solving vector equation

Suppose I know the following 3 normalized vectors in 3-D Euclidean space: $\frac{\vec{A}}{||\vec{A}||}$, $\frac{ \vec{A} + \vec{B}}{||\vec{A} + \vec{B}||}$ $\frac{ \vec{A} + \vec{C}}{||\vec{A} + ...
0
votes
3answers
171 views

What do I call a unit vector parallel to a coordinate axis?

What do I call an arbitrary element of this set of vectors? $$ \begin{align*} \{&\langle 1, 0, 0 \rangle, \\ &\langle 0, 1, 0 \rangle, \\ &\langle 0, 0, 1 \rangle, \\ &\langle ...
2
votes
1answer
86 views

Prove that $\vec{EA}+\vec{EB}+\vec{EC}+\vec{ED}=2\vec{EO}$

Let there be a circle $(O,R)$ and $AB,CD$ two perpendicular chords of that circle that intersect on point $E$. Prove that $\vec{EA}+\vec{EB}+\vec{EC}+\vec{ED}=2\vec{EO}$