1
vote
2answers
45 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 ...
5
votes
0answers
51 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
70 views

Cross product, ortonormal basis

Could you explain to me why for $\{i, \ j, \ k\}$ an orthonormal basis of $\mathbb{R}^3$ we have $i \times j =k, \ \ j \times k = i, \ \ k \times i =j$? Thank you.
1
vote
1answer
44 views

Comparing a geometric definition of cross product to the “usual” one

Could you help me with my little problem? Given this definition of cross product: 1) $a \times b$ is perpendicular to $a$ and $b$, whenever $ a,b$ are linearly independent 2) basis $a, \ b, \ a ...
1
vote
3answers
8k views

Area of a parallelogram, vertices $(-1,-1), (4,1), (5,3), (10,5)$.

I need to find the area of a parallelogram with vertices $(-1,-1), (4,1), (5,3), (10,5)$.
8
votes
1answer
190 views

Maps of $\mathbb{R}^3$ preserving the cross product

Given a map $\phi:\Bbb R^3 \rightarrow \Bbb R^3$ such that for all $a,b \in \Bbb R^3$: $$\phi(a \times b)=\phi(a) \times \phi(b)$$ Is $\phi$ necessarily a rotation around the origin or the map ...
2
votes
0answers
195 views

Cross product in higher than 3 dimensions

As I understand it, to get an $n$-dimensional cross product, you need $n-1$ vectors of dimension $n$. However my lecture notes are quite miss leading in the fact that they suggest this isn't always ...
3
votes
2answers
1k views

How do you compute the normal vector to a hyperplane in $\mathbb{R}^n$ given $n$ representative points?

Given $n$ points (no two identical, no three colinear, no four coplanar, etc.), I'd like to find a formula for the normal vector to the unique hyperplane that intersects each of these points. In ...
4
votes
6answers
3k views

Visual Ways to Remember Cross products of Unit vectors? Cross-product in $\mathbb F^3$?

Objective to find visual and accessible ways to remember this formula fast $$(x,y,z)\times(u,v,w)=(yw-zv,zu-xw,xv-yu)$$ I have used Sarrus' rule but it is slow, more here. Since it is slow, I have ...