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.

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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 $\vec b$,…
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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…
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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…
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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…
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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
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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…
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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…
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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…
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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 $b$?
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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
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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?
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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
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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…
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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…
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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: Why is it not possible to define a cross product for other…
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