# 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.

1,121 questions
Filter by
Sorted by
Tagged with
24 views

### Skew Symmetric Matrix vs. Cross Product

This might be more of a programming question in truth (at least, I suspect the answer is related to computer programming), but I figured I'd ask here. Why would someone choose to represent a cross ...
1 vote
51 views

### Understanding the Relationship Between Cross Product Components and Differential Forms on a Membrane

I am struggling with a differential geometry and vector calculus concept involving the relationship between the cross-product components and differential forms. The specific context is a response ...
• 441
42 views

### Help with proof of Curl Double Product identity using Geometric Algebra. Most things seem to fall in place, but having a few issues.

So I'm pretty new to GA/Clifford Algebras, but it's been fairly interesting so far. I figured I'd try to prove some basic vector calculus identities with it, just to help me get my bearings. I decided ...
• 389
22 views

### Surface area element in 4-dimensions

In Dirac's "General Theory of Relativity" (p. 40) he says "If we take two small contravariant vectors $\xi^\mu$ and $\zeta^\mu$, the element of surface area that they subtend is ...
41 views

### Loop integral of $\mathbf{r}\times d\mathbf{r}$ is equal to twice the enclosed area?

I came across the following integral in the literature for a loop $l$ that goes around a surface of area $A$: $$\oint_l \mathbf{r}\times d\mathbf{r} = 2 A \mathbf{\hat{n}}$$ where $\mathbf{\hat{n}}$ ...
• 103
34 views

### Tangent Vector, Principal normal vector, Binormal vector, and Torsion

So I'm trying to fully grasp how all these relate. My current understanding is that the tangent vector describes the direction in which the curve is going/curving. Meanwhile, the principal norm is ...
17 views

### Cross product properties.

I just want to know if it's correct: let $\vec{v}=(\vec{a}+\alpha\vec{b})\times(2\vec{a}+\vec{b})$, with $\alpha\in\mathbb{R}$. If $||\vec{a}||=\sqrt{2},||\vec{b}||=1$ and the angle between $\vec{a}$ ...
• 2,075
32 views

### What is a space which is the cross product $D^2\times S^1$?

While taking cross product of two one-spheres, $S^1\times S^1$ seems esasy to imagine/identify with a torus $T^2$, I struggle to make a picture of a $D^2\times S^1$, what kind of a space is that in ...
• 339
37 views

• 663
89 views

### Computing $U \times V$

Problem: Given $$U = (2,1,3)$$ $$U = (4,-2,2)$$ find $U \times V$. Answer: \begin{align*} U \times V &= \begin{vmatrix} i & j & k \\ 2 & 1 & 3 \\ 4 & -2 & 2 \end{...
• 3,994
1 vote
128 views

• 2,109
69 views

### Is it true that $\vec{\nabla}\times[(\vec{a}\cdot\vec{\nabla})\vec{a}]=\vec{\nabla}\times[\vec{a}\times(\vec{\nabla}\times\vec{a})]$?

I'm reading some notes on particle physics by a university professor, and after doing calculations I've reached the conclusion that in order for one of his claims to be true, this equality would have ...
• 643
86 views

### Prove a subset $E$ of $\mathbb{R}^3$ is exactly a plane if there are vectors $u,v,w\in\mathbb{R}^3$ such that $v$ and $w$ are linearly independent and

Prove that a subset $E$ of $\mathbb{R}^3$ is exactly a plane, if there exists vectors $u,v,w \in \mathbb{R}^3$, such that $v$ and $w$ are linearly independent and $$E = u + \mathbb{R}v + \mathbb{R}w$$ ...
• 1,135
505 views

• 8,720
35 views

78 views

### Can we define this alternate version of the cross product?

I've been playing with the definition of the cross product and am trying to grasp the atomic algebraic assumptions needed to define the unique cross product. I remember seeing a post that was saying ...
• 1,105
34 views

• 4,412
83 views

### Identity for a scalar quintuple product?

I find myself needing to cross two pairs of vectors, and cross that result (so a normal of normals) and check whether each of two of the original points are on different sides of the plane it defines: ...
• 583