Questions tagged [cross-product]

In $\Bbb R^3$, the cross product of two vectors $v$ and $w$ produces a vector $v \times w$ perpendicular to both. This tag is not meant for products in other mathematical contexts, such as products of groups (such as the [tag:direct-product]), sets (the Cartesian product), graphs, and so on.

In mathematics, the cross product, vector product, or Gibbs' vector product, is a binary operation on two vectors in $3$-dimensional space. It results in a vector which is perpendicular to both of the vectors being multiplied, and therefore normal to the plane containing them.

We write:

$$\vec{u}\times\vec{v}=\begin{pmatrix}u_1\\ u_2\\ u_3\end{pmatrix}\times\begin{pmatrix}v_1 \\ v_2 \\ v_3\end{pmatrix}=\begin{pmatrix}u_2v_3-u_3v_2 \\ u_3v_1-u_1v_3 \\ u_1v_2-u_2v_1\end{pmatrix} $$

The cross product is anti-commutative, so $\vec{u}\times\vec{v}=-(\vec{v}\times\vec{u})$.

The norm of the cross-product has several important geometric properties. For instance, $||\vec{u}\times \vec{v}||$ is the area of the parallelogram spanned by $\vec{u},\vec{v}$, i.e. $||\vec{u}\times \vec{v}||=||\vec{u}||||\vec{v}||\sin(\theta)$ where $\theta$ is the angle between them. This in turn yields Lagrange's identity $$ ||\vec{u}\times \vec{v}||^2 + |\vec{u}\cdot\vec{v}|^2 = (||\vec{u}||||\vec{v}||)^2 $$If $\vec{u},\vec{v},\vec{w}\in\mathbb{R}^3$, the volume of the parallelipiped spanned by them is $|(\vec{u}\times\vec{v})\cdot \vec{w}|$.

There is also a seven-dimensional cross product as bilinear operation on vectors in seven dimensional Euclidean space.

1026 questions
98
votes
7 answers

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 to the plane made by $\vec a$ and $\vec b$,…
VF1
  • 1,791
  • 1
  • 14
  • 21
69
votes
1 answer

What is the logic/rationale behind the vector cross product?

I don't think I ever understood the rationale behind this. I get that the dot product $\mathbf{a} \cdot \mathbf{b} =\lVert \mathbf{a}\rVert \cdot\lVert \mathbf{b}\rVert \cos\theta$ is derived from the cosine rule. (Do correct me if I'm…
Danxe
  • 1,635
  • 1
  • 14
  • 24
60
votes
6 answers

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 workings of them... I could get the cross product…
Curiosity
  • 1,388
  • 2
  • 14
  • 16
56
votes
13 answers

Why is cross product defined in the way that it is?

$\mathbf{a}\times \mathbf{b}$ follows the right hand rule? Why not left hand rule? Why is it $a b \sin (x)$ times the perpendicular vector? Why is $\sin (x)$ used with the vectors but $\cos(x)$ is a scalar product? So why is cross product defined…
koe
  • 737
  • 7
  • 11
53
votes
4 answers

Understanding Dot and Cross Product

What purposes do the Dot and Cross products serve? Do you have any clear examples of when you would use them?
David McGraw
  • 865
  • 2
  • 13
  • 11
40
votes
5 answers

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 two vectors defined (in Differential Geometry of Curves and Surfaces by Manfredo Perdigão do Carmo) by: Let $\mathbf{u}$, $\mathbf{v}$ be in…
36
votes
4 answers

Cross product in higher dimensions

Suppose we have a vector $(a,b)$ in $2$-space. Then the vector $(-b,a)$ is orthogonal to the one we started with. Furthermore, the function $$(a,b) \mapsto (-b,a)$$ is linear. Suppose instead we have two vectors $x$ and $y$ in $3$-space. Then the…
34
votes
8 answers

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 $\textbf{n} = \textbf{a} \times \textbf{b}$ give me a vector which is perpendicular to a…
jjepsuomi
  • 8,135
  • 12
  • 49
  • 89
33
votes
5 answers

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 $b$?
user7087
28
votes
2 answers

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

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 mysterious to me.
William Chang
  • 2,229
  • 1
  • 18
  • 28
27
votes
3 answers

Is a function that preserves the cross product necessarily linear in $\mathbb R^3$? $f(a) \times f(b) = a \times b$

Assume that $f: \mathbb{R^3} \rightarrow \mathbb{R^3}$ is a function such that $$ f(a) \times f(b)=a \times b $$ for all $a,b \in \mathbb{R^3}$, where ''$\times$'' denotes the cross product in $\mathbb{R^3}$. Does $f$ have to be a linear mapping?
Alex
  • 2,111
  • 16
  • 25
26
votes
2 answers

Is the cross product of two unit vectors itself a unit vector?

Or, in general, what does the magnitude of the cross product mean? How would you prove or disprove this?
May Oakes
  • 403
  • 1
  • 4
  • 6
24
votes
1 answer

Is the seven-dimensional cross product unique?

I'm confused about how many different 7D cross products there are. I'm defining a 7D cross product to be any bilinear map $V \times V \to V$ (where $V$ is the inner product space $\mathbb{R}^7$ endowed with the Euclidean inner product) such that for…
22
votes
1 answer

Generalized Cross Product

I know that the cross product can be generalized as $$\text{cross}(x_0,...,x_{n-1})=\det\begin{vmatrix}&x_0&\\&x_1&\\&\vdots&\\e_1&\cdots&e_n\end{vmatrix}$$ where $e_i$ is the $i$'th standard unit vector. We have $n-1$ vectors in $n$-dimensional…
user142299
20
votes
4 answers

Cross product in $\mathbb R^n$

I read that the cross product can't be generalized to $\mathbb R^n$. Then I found that in $n=7$ there is a Cross product: https://en.wikipedia.org/wiki/Seven-dimensional_cross_product Why is it not possible to define a cross product for other…
Anna
  • 1,617
  • 2
  • 14
  • 20
1
2 3
68 69