For questions on cross products.

learn more… | top users | synonyms

0
votes
1answer
54 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. $i^2=j^2=k^2=ijk=-1$, ...
1
vote
1answer
47 views

What is the difference between $|a|$ and $|| a ||$

So I am doing maths involving cross-product and dot-product and I came across the above two notation as in $||u\times v|| = ||u|| ||v|| \sin a$ and $u\cdot v = |u| |v| \cos a$. What is the difference ...
3
votes
1answer
186 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 ...
3
votes
0answers
25 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 ...
0
votes
2answers
44 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 ...
0
votes
2answers
46 views

How to find a normal vector from an equation in the form f(x,y)?

If I have an equation $f(x,y)$ which given the $x$ and $y$ coordinate, it gives you the $z$ coordinate. How can I find the normal (directional) vector of the the point $(x,y,f(x,y))$? This would be ...
3
votes
1answer
57 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
2answers
58 views

Adding two vectors such that the resulting vector is perpendicular to a third vector

Let $$a = (-3, 3, 1)$$ $$b = (1, 4, -4)$$ $$c = (2, 1, -3)$$ For which values of $t \in \Re$ is $b + tc$ perpendicular to a? For a vector to be perpendicular to $a$, the dot product of that ...
1
vote
2answers
133 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
0answers
10 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 ...
11
votes
2answers
1k views

Why is cross product only defined in 3 and 7 dimensions? [duplicate]

Why $3$ and $7$? I know from some reading that Hurwitz's Theorem explains this, but can someone help me build some intuition behind this or perhaps provide a simpler explanation? It still seems ...
1
vote
1answer
26 views

cross-products versus units of measure

If I draw 2 perpendicular line segments on the ground, 3 meters and 4 meters, how far into the sky does their cross-product extend? What if instead the line lengths are 300 cm and 400 cm? Can ...
0
votes
1answer
58 views

Prove that $g(t) \times \frac{d}{dt} g(t) = 0$

If the vectorial function $r = g(t)$, with values in $\mathbb R^3$ and where $t\in\mathbb R$, is a solution of the differential equation $\frac{d^2}{dt^2} r(t) = t^2 r(t)$, such that $g(0) = 0$, ...
4
votes
1answer
93 views

Can cross products be defined without coordinates?

I recently learned about cross products and understood that cross products can be computed without an origin and coordinates in three dimensions, like vectors can be defined without coordinates. But ...
1
vote
0answers
37 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 $13x+y-5z = 0$?
3
votes
1answer
71 views

Cross product and right hand rule

Is there a simple proof that the cross product (defined as the usual determinant) always obeys the right hand rule?
0
votes
1answer
85 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 ...
4
votes
2answers
114 views

Cross product in > 3d

What exactly would we get by calculating the cross product of vectors in $R^n, n>3$ using the formula $\vec a \times\vec b=(||\vec a||||\vec b||\sin\Theta)\vec n$ $\vec n$ being a vector normal ...
0
votes
2answers
63 views

Vector triple product = 0

Let $U, V, W$ be three non-zero vectors, no two of which are parallel. Under what conditions is $U\times(V\times W) = 0$?
2
votes
1answer
91 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 ...
0
votes
0answers
103 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}{=} ...
1
vote
2answers
1k views

Equation of plane that goes for intersection of 2 planes and is perpindicular to another plane

Really don't know what to do here, went to a tutor neither did he. Okay first the problem: Find the equation of the plane that passes through the line of intersection of the planes x − z = 2 and y + ...
2
votes
1answer
47 views

Relation between $\vec{a}\times\left(\vec{b}\times\vec{c}\right)$ and $\left(\vec{a}\times\vec{b}\right)\times\vec{c}$

The operation $\vec{a}\times\left(\vec{b}\times\vec{c}\right)$ can be simplified to $\vec{b}\left(\vec{a}\cdot\vec{c}\right) - \vec{c}\left(\vec{a}\cdot\vec{b}\right)$ and can easily be remembered by ...
1
vote
0answers
39 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 ...
0
votes
2answers
49 views

Prove that vectors x,y are linearly dependent exactly when …

Prove that vectors $\vec{x},\vec{y}$ (belonging to $\mathbb{R}^3$) are linearly dependent only if the following is true $$ \begin{vmatrix} x_1&y_1 \\ x_2&y_2 \end{vmatrix} ...
4
votes
0answers
173 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 ...
0
votes
2answers
496 views

Finding the volume of a pyramid (the vector way)

The problem I have 4 points $ P \; (-1,2,0) \\ Q \; (2,1,3) \\ R \; (1,0,1) \\ S \; (3,-2,3) $ and I want to find the volume of a pyramid. What I'm most concerned here is the appropriate strategy ...
1
vote
1answer
161 views

Projective geometry. Interpretation of a cross product between a line coincident with a point

Let $p \in \mathcal{P}^2$ be a point in projective 2-space coincident with a line $l\in\mathcal{P}^2$ such that $l^\top p = 0$. What does $l \times p$ mean? For example, $p = ...
2
votes
2answers
235 views

How to find all 3 orthogonal vectors to a 4D vector

For a program I'm writing, I need to find the vectors orthogonal to a given vector rotated at an arbitrary angle, and in 4D. It is a unit vector. For 3D, I found the two orthogonal vectors like ...
2
votes
5answers
99 views

Reasoning behind the cross products used to find area

Alright, so I do not have any issues with calculating the area between two vectors. That part is easy. Everywhere that I looked seemed to explain how to calculate the area, but not why the cross ...
1
vote
0answers
29 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
1answer
263 views

Finding 2 vectors orthogonal to each other and a given vector

given vector: $u = <1, -7, 2>$ Find a vector $v$ and $w$ which are are orthogonal to each other and to $u$ I tried the following: $$ v \cdot u = <1,1,c> \cdot <1,-7,2> \rightarrow ...
0
votes
2answers
298 views

Proof in a Scalar Triple Product [closed]

For any three vectors $\vec a,\ \vec b,\ \vec c$, show that : $$[\vec a\times\vec b,\ \vec b\times\vec c,\ \vec c\times\vec a]=[\vec a,\ \vec b,\ \vec c]^2$$ where $[\vec a,\vec b,\vec c]=\vec ...
5
votes
0answers
54 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 ...
0
votes
1answer
38 views

Given 3 cross products of 3 vectors, how do you solve an expression of this format?

If you're given: $$a \times b = (2, -4, 2),\quad a \times c = (7, 13, -11),\quad b \times c = (1, 7, 1)$$ what properties of cross products or formulas can you use to solve $(2b - c) \times (3a + ...
0
votes
0answers
39 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 ...
0
votes
1answer
62 views

Projection on a hyperplan and a hypercube intersection

I need to project an array y onto a hyperspace defined by (a.x) = c where a is an array in R^N However, x needs to belong in the hypercube {0 <= x_i <= 1, for all i from 1 to n} Therefore from ...
0
votes
3answers
84 views

Cross product of vector functions

I was trying to make sense of a problem when I stumbled upon this on yahoo answers. I was just wondering if it was correct. If it is, can you please maybe explain why? ${\bf r}'(t) = \langle -5 \cos ...
2
votes
1answer
60 views

Why and how are quaternions 'bilinear'?

What does it mean when we say that quaternion composition is 'bilinear'? I have observed that some authors write quaternion multiplication as: While others specify: Excuse the poor images, ...
2
votes
1answer
193 views

Showing that a set of points equidistant to two other points form a plane.

Question: if p and q are two distinct points in space, show that the set of points equidistant from p and q form a plane. Work Done: Note: I'm pretty sure this can be done with vectors and cross ...
1
vote
3answers
604 views

How come the cross product of two planes is collinear with the direction vector of the line?

If two planes intersect in a line, explain why the cross product of the normal vectors of the planes is collinear with the direction vector of the line.
3
votes
1answer
101 views

Show that $e^{\theta(s\times)} = I + \sin\theta(s\times) + (1 − \cos \theta)(s\times)^2$

$$e^{\theta(s\times)} = I + \sin\theta(s\times) + (1 − \cos \theta)(s\times)^2$$ I have to prove the above formula and am not sure where to start, may someone please help me! The full question is ...
0
votes
2answers
62 views

Show that angular momentum can be divided into center of mass and internal coordinate

This might belong in physics but I want to be sure I am approaching the math right. Given: show angular momentum can be divided into separate parts of the center of mass and internal coordinates. ...
2
votes
1answer
48 views

Explanation for cross product observation

For a dynamics class, I have observed a weird correlation which my Professor couldn't explain. So I was wondering if someone would give me an explanation of what's happening . Here's a general ...
1
vote
3answers
123 views

How do I plot a bunch of vectors in Maple and find the difference and cross product between them?

Title says it all. This is the assignment I'm trying to do. http://math.rutgers.edu/~shtelen/Teaching/Fall-2013/L1_instr.pdf Data I need to plot ...
1
vote
1answer
164 views

on proving bac-cab rule, $\vec{A} \times ( \vec{B} \times \vec{C})= \vec{B} (\vec{A} \cdot\vec{C})- \vec{C}(\vec{A}\cdot\vec{B})$

I noticed something when I was doing a proof of the BAC-CAB rule, and wanted to check if my intuition was correct. First, when I actually multiplied out $\vec{B} (\vec{A} ...
2
votes
3answers
551 views

showing / proving curl identity $\nabla \times \left( \frac{1}{r^2} \hat r \right) = 0$

OK, I have to show the following: $$ \nabla \times \left( \frac{1}{r^2} \hat r \right) = 0$$ This should be pretty easy, but I wanted to be sure I was doing this correctly. I set up the matrix: ...
0
votes
3answers
56 views

Help proving that $\vec{a}\times(\vec{a}\times(\vec{a}\times\vec{b}))\cdot\vec{c} = -\|\vec{a}\|^2\vec{a}\cdot\vec{b}\times\vec{c}$

This is problem 13 from Chapter 13, Section 14 of Apostol's Calculus Vol 1. I need to prove or disprove the formula $\vec{a}\times(\vec{a}\times(\vec{a}\times\vec{b}))\cdot\vec{c} = ...
2
votes
1answer
43 views

If $U \times V$ is a unit vector, $|U| = \sqrt{3}/3, |V| = 2$ and $U\cdot V > 0$, then what is the angle between $U$ and $V$?

If $U \times V$ is a unit vector, $|U| = \sqrt{3}/3$, $|V| = 2$ and $U \cdot V > 0$, then what is the angle between $U$ and $V$? Is this just dot product? $U \cdot V = |U| |V| \cos(\theta)$? Solve ...
2
votes
1answer
48 views

Unbiased estimate of cross-product for unbiased vector

Let $g$ be an unbiased estimate of a vector $G$. Can $g$ be used to find an unbiased estimate of the cross product $GG'$? I'm stuck because naively using $gg'$ is a biased estimator, with the ...