For questions on cross products.

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3
votes
3answers
150 views

Cross product- square

I recently saw the following expression somewhere- $$\frac{1}{2} \left\| \frac{\vec{u}}{9} \times \frac{\vec{u} + \vec{v}}{9} \right\| + \frac{1}{2} \left\| \frac{\vec{u} + \vec{v}}{9}\times ...
1
vote
2answers
66 views

Line integrals, cross products, surface integrals and Stoke's Theorem related problem?

The vector field $\vec{F}(\vec{R})$ is defined as being equal to the line integral over some simple closed curve $C$: $$\vec{F}(\vec{R})=\oint_C\|\vec{r}-\vec{R}\|^2d\vec{r}.$$ We show that there ...
0
votes
2answers
38 views

About matrix $R$, what is this called: $R^TR$? What is it for?

I am doing singular value decomposition on a matrix $R$. The first step is to compute such a matrix $R^TR$. What is this matrix? A reference told me this is cross product of matrix R. I use a ...
-1
votes
2answers
521 views

Find the sine sign given a pair of 3D vectors

I want to find the exact sine between two vectors in 3-dimensional space. Data: $x$: vector $y$: vector $z = \Vert x \times y \Vert$ I have tried this: $$\sin \alpha = \frac{\Vert z\Vert}{( ...
4
votes
1answer
87 views

How to solve cross-products including matrices?

I'm a programmer and I'm doing a whitebalance-transformation in RGB colorspace. This should work with this transformation matrix that I've found in literature: $$ \begin{pmatrix} R \\ G \\ B ...
2
votes
1answer
33 views

How do you find the max value of a length of a vector?

I have a vector $v = 7j$ and a vector $u$ with a length of 5 that starts at the origin and rotates in the $xy$-plane. How am I supposed to find the max value of the length of the vector $|u \times ...
1
vote
1answer
20 views

Cross Product in Levi-Civita Notation - The elementary basis vector's missing?

http://www.unl.edu.ar/ceneha/uploads/Cartesian_tensors_Index_notation_&_summation_convention.pdf avers: $1.$ $(a×b).(c×d) = \epsilon_{i jk}a_jb_k \quad e_{ilm}c_ld_m$ $2. \nabla × ...
0
votes
1answer
18 views

Motivation for construction of cross-product (Quaternions?)

I just found a very interesting article here: http://www.johndcook.com/blog/2012/02/15/dot-cross-and-quaternion-products/ The author observes that by defining i,j,k s.t. $i^2=j^2=k^2=ijk=-1$, ...
0
votes
1answer
33 views

Find the length and direction of $u \times v$ and $v \times u$

So I was given two vectors: $u=-8i- 2j- 4k$, and $v=2i+2j+k$. I was able to figure out the cross product of $u\times v$ which is $6i-12k$, and $v \times u$ which is $-6i+12k$. However, I need help ...
5
votes
0answers
37 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 ...
4
votes
0answers
122 views

Cross Product - Moments :: Dynamics

Some background: I am self studying dynamics and I have encountered a fundamental problem with either my understanding of linear algebra, or I am just plain dumb. So, I print screened the page of the ...
3
votes
0answers
17 views

Higher Dimensional Right-Hand Rule

In seven dimensions, the cross product makes sense. Without resorting to nonvector tensors or exterior products (although they can be used to further explain), how does one perform this cross product ...
2
votes
0answers
190 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 ...
1
vote
0answers
21 views

Name of an identity for traceless matrices in $\mathbb{R}^3$?

While working on a more compact presentation of a derivation in the context of incompressible fluid flow we tried to simplify things by introducing intermediate steps instead of writing out lengthy ...
1
vote
0answers
34 views

Given two lines, how do I find the plane?

$$r_1(t) = \langle t, 2t, 3t\rangle$$ $$r_2(t) = \langle3t, t, 8t\rangle$$ I found $\mathbf{n} = \langle13,1,-5\rangle$ Can I just plug in say $P_0 = (0,0,0)$ and get $13x+y-5z = 0$?
1
vote
0answers
35 views

Cross Product Component Values

When taking the cross product, the x component of the perpendicular vector is the (signed) area of the yz projection of the parallelogram spanned by the two vectors it's orthogonal to-right? And ...
1
vote
0answers
21 views

Matrix: Area of a Triangle, which point to choose for cross multiplication

When given 3 points(vertices), which one should you pick to do the your calculations with. E.g.: $P1=(1,-1,1) P2=(2,1,-1) P3=(1,-2,-1) $ I can pick P1 -> P2 and P1 -> P3. Then do my cross ...
0
votes
0answers
17 views

How to prove the distributive law of cross product by geometric definition

I'm asked to prove the distributive law of cross product by its geometric definition in an exercise, and I found the following answer in stack exchange. How to prove the distributive property of ...
0
votes
0answers
11 views

Find an equation of the plane through the line of intersection…

Find an equation of the plane through the line of intersection of the planes x - z = 1 and y + 2z = 3 and perpendicular to the plane x + y - 2z = 1. Is this set up right? x y z 1 0 ...
0
votes
0answers
9 views

Cross Product of Covectors

Is the vector/cross product defined for covectors (in the dual space) or is it, strictly speaking, only defined for vectors themselves? I would imagine that it works fine for covectors but I wanted to ...
0
votes
0answers
102 views

What is does the transformation $[\mathbf{a}]_{\times}$ do?

https://en.wikipedia.org/wiki/Cross_product#Conversion_to_matrix_multiplication I'm curious about the matrix $[\mathbf{a}]_{\times} \stackrel{\rm def}{=} ...
0
votes
0answers
43 views

How do I prove that there is a minus in the vector product?

It's basically this problem but it wasn't welcomed in [academia.SE] . My teacher says this is the way to obtain the vector product of two vectors: $$\overrightarrow{u} \times \overrightarrow{v} = ...
0
votes
0answers
32 views

Given the normal vector n(s), determines the curvature k(s) and the torsion

Given the normal vector n(s) of a curve $\alpha$, with non zero torsion everywhere, determines the curvature k(s) and the torsion $\tau$(s) of $\alpha$. I am first trying to show the following which ...
0
votes
0answers
33 views

Cross-Product Intuition

Can anyone provide me with a little geometric insight as to why the ratio of the vector components of some vector perpendicular to two other should simply be the projections of those two vectors onto ...