For questions on cross products.

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19
votes
6answers
4k views

Is the vector cross product only defined for 3D?

Wikipedia introduces the vector product for two vectors $\vec a$ and $\vec b$ as $$ \vec a \times\vec b=(||\vec a||||\vec b||\sin\Theta)\vec n $$ It then mentions that $\vec n$ is the vector normal ...
16
votes
4answers
3k views

Origin of the dot and cross product?

Most questions usually just relate to what these can be used for, that's fairly obvious to me since I've been programming 3D games/simulations for a while, but I've never really understood the inner ...
14
votes
1answer
996 views

Wedge Product, A Novel Interpretation or Just Plain Wrong?

I have read (I think) all of the previous threads on this website (and many others) on this topic & unfortunately have not found an answer to my question. Due to the fact that I am only beginning ...
11
votes
4answers
4k views

Wedge product and cross product - any difference?

I'm taking a course in differential geometry, and have here been introduced to the wedge product of to vectors defined (in Differential Geometry of Curves and Surfaces by Manfredo Perdigão do Carmo) ...
10
votes
2answers
900 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 ...
8
votes
5answers
3k views

Why does cross product give a vector which is perpendicular to a plane

I was wondering if anyone could give me the intuition behind the cross product of two vectors $\textbf{a}$ and $\textbf{b}$. Why does their cross product $\textbf{n} = \textbf{a} \times \textbf{b}$ ...
8
votes
1answer
181 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 ...
7
votes
2answers
2k views

What's the opposite of a cross product?

For example, $a \times b = c$ If you only know $a$ and $c$, what method can you use to find $b$?
7
votes
1answer
246 views

Does this cross-product norm inequality hold?

Let $\times$ denote the cross-product. $\;$ Is it the case that For all unit vectors $\:\mathbf{x}\hspace{.01 in},\hspace{-0.03 in}\mathbf{y}\hspace{-0.03 in},\hspace{-0.02 in}\mathbf{z}\:$ in ...
6
votes
5answers
299 views

Sphere equation given 4 points

Find the equation of the Sphere give the 4 points (3,2,1), (1,-2,-3), (2,1,3) and (-1,1,2). The *failed* solution I tried is kinda straigh forward: We need to find the center of the sphere. Having ...
6
votes
2answers
145 views

Explanation of a cross product result

In my book the result $$(u\times v)\cdot(x\times y)=\begin{vmatrix} u\cdot x & v\cdot x \\u \cdot y & v \cdot y\end{vmatrix},$$ where u, v, x and y are arbitrary vectors, is stated (here ...
5
votes
1answer
447 views

Do the BAC-CAB identity for triple vector product have some intepretation?

As in the title, I was wondering if the formula: $$a\times (b\times c)=b(a\cdot c)-c(a \cdot b)$$ for $\mathbb R ^3$ cross product has some geometrical interpretation. I've recently seen a proof (from ...
5
votes
3answers
96 views

How to divide by $(a_1,a_2,a_3)$

I have been searching for an explanation in Howard's Linear Algebra and couldn't find an identical example to the one below. The example tells me that vectors $\boldsymbol{a}_1$, $\boldsymbol{a}_2$ ...
5
votes
1answer
774 views

Cross product of vectors as a determinant: valid matrix operation?

"The definition of the cross product can also be represented by the determinant of a formal matrix." —Wikipedia This seems like a hack to me—something of much practical use but ...
5
votes
1answer
139 views

The proof of $\hat{b}(\hat{a}\cdot\hat{c})-\hat{c}(\hat{a}\cdot\hat{b})=\hat{a}\times(\hat{b}\times\hat{c})$

formula: $\hat{b}(\hat{a}\cdot\hat{c})-\hat{c}(\hat{a}\cdot\hat{b})=\hat{a}\times(\hat{b}\times\hat{c})$ $\hat{a}\times(\hat{b}\times\hat{c})$ is on the $\hat{b}$, $\hat{c}$ plane, so: ...
5
votes
0answers
42 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
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 ...
4
votes
3answers
310 views

Rotational invariance of cross product

Hi guys I'm looking for a proof that $ ( Ra \times Rb ) = R ( a \times b ) $ where $\times$ is the three-dimensional cross product, and $R$ is a rotational matrix ( $\det R = 1$ and $R^T R = I$ ) ...
4
votes
2answers
541 views

Example of an associative cross product, any significance?

While trying to find cases that showed the cross product is not associative, I found some that were. I'm trying to show that $(\mathbf{A}\times \mathbf{B}) \times \mathbf{C} \ne \mathbf{A}\times ...
4
votes
2answers
84 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 ...
4
votes
1answer
82 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 ...
4
votes
2answers
136 views

Why is $\det(\vec{A},\vec{B}) = |\vec{A} \times \vec{B}|$?

In the multivariable calculus class the teacher showed us the formula of the cross product $$ \vec{A} \times \vec{B} =\begin{vmatrix}\hat{\imath}& \hat{\jmath}& \hat{k} \\ a_1 & a_2 ...
4
votes
1answer
453 views

Interpretation of eigenvectors of cross product

If we fix a non-zero vector $\boldsymbol{v}\in\mathbb{R}^3$, then the linear map $\boldsymbol{x}\mapsto\boldsymbol{v}\times\boldsymbol{x}$ has trivial eigenvectors $\boldsymbol{x}_1=t\boldsymbol{v}$ ...
4
votes
0answers
136 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 ...
4
votes
1answer
89 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 ...
3
votes
4answers
8k views

What is the general formula for calculating dot and cross products in spherical coordinates?

I was writing a C++ class for working with 3D vectors. I have written operations in the Cartesian coordinates easily, but I'm stuck and very confused at spherical coordinates. I googled my question ...
3
votes
2answers
62 views

Are these two rotation matrices related?

There are two $3\times3$ matrices $A$ and $B.$ Both represent a rotation in 3D space. $A$ and $B$ are given as follows where $a,b,c$ are column vectors. $A = [\begin{array}{ccc}a & b & c \\ ...
3
votes
2answers
173 views

Help over the proof of triple vector product identity

For all vectors $\bf{x}$, $\bf{y}$ and $\bf{z}$, $$\bf{x}\times(\bf{y}\times\bf{z})=(\bf{x}\cdot\bf{z})\bf{y}-(\bf{x}\cdot\bf{y})\bf{z}$$ The proof goes as follows: We may suppose that $\bf{y}$ ...
3
votes
5answers
515 views

Help understanding cross-product

I am trying to calculate the intersection point (if any) of two line segments for a 2D computer game. I am trying to use this method, but I want to make sure I understand what is going on as I do it. ...
3
votes
1answer
54 views

cross product in cylindrical coordinates

Hi i know this is a really really simple question but it has me confused. I want to calculate the cross product of two vectors $$ \vec a \times \vec r. $$ The vectors are given by $$ \vec a= a\hat ...
3
votes
1answer
178 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
1answer
53 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?
3
votes
1answer
99 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 ...
3
votes
1answer
51 views

Proof that $\mathbf{R}[\omega]_\times\mathbf{R} = [\mathbf{R}\omega]_\times$

I have to prove that $$\mathbf{R}[\omega]_\times\mathbf{R}^\mathrm{T} = [\mathbf{R}\omega]_\times$$ Herein $\omega$ is a vector with elements. The notation $[\mathbf{a}]_\times$ is a conversion of ...
3
votes
1answer
405 views

Determining partial derivatives and cross products for bicubic interpolation using function values only?

I'm trying to implement a bicubic interpolation algorithm. In order to calculate the interpolated values, I need to calculate sixteen coefficients used in the calculation process - and that's where ...
3
votes
1answer
56 views

Is the matrix form of the cross product related to bilinear forms.

The cross product of two vectors $\mathbf{x}, \mathbf{y} \in \mathbb{R}^3$ can be represented as a matrix product as follows, if $\mathbf{x} = (x_1, x_2, x_3)^{\top}$ then $\mathbf{x} \times ...
3
votes
0answers
19 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 ...
3
votes
1answer
38 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 × ...
3
votes
3answers
171 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 ...
2
votes
1answer
795 views

invariance of cross product under coordinates rotation

Question goes as If $\vec A$ and $\vec B$ are invariant under rotation, the prove that $ \vec A \times \vec B $ is also invariant. However solution of on the other page is not given. Says ...
2
votes
1answer
137 views

How do I solve $F = \nabla\times G$ for $G$?

Given the vector valued function $F(x,y,z) = (xz,-yz,y)$ find $G$ such that $F = \nabla\times G$ I let $G(x,y,z) = (G_1,G_2,G_3)$ and expanded $\nabla \times G$ then equated the components to $F$ but ...
2
votes
1answer
45 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 ...
2
votes
5answers
94 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 ...
2
votes
1answer
68 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.
2
votes
2answers
226 views

How to prove the equality of two vectors?

OK, i am trying to prove that if $\vec a\times \vec b = \vec a \times \vec c$ and also $\vec a\cdot \vec b = \vec a \cdot \vec c$ then $\vec b = \vec c$. so far i got to $\vec n \tan \alpha = \vec m ...
2
votes
2answers
61 views

Cocountable fibers

Let $C$ be an uncountable set. Can we construct a set $A \subseteq C^2$ such that it has a cocountable number of cocountable horizontal fibers, and a cocountable number of countable vertical fibers?
2
votes
2answers
157 views

Vectors question

I'm trying to prove whether the followings statements are true or not. I would appreciate your help, as I'm not sure how to begin. Given: $ u,x_n \in \mathbb{R}^3$ and for every $n$, let $x_{n+1}=u ...
2
votes
3answers
449 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: ...
2
votes
2answers
971 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 ...
2
votes
1answer
2k views

How do you integrate Cross Products?

Hey I'm doing a course in mechanics and these keep cropping up! So for this question I'm working in 3d, and so far have $$m \mathbf{k} \cdot (\mathbf{q} \times \ddot{\mathbf{q}} )=0$$ so I need ...