# 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.

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### What is the cross product

I know that the cross product of 2 points $(1,2)$ and $(a,b,c)$ is $\{(1,a),(1,b),(1,c),(2,a),(2,b),(2,c)\}$ But what is the cross product of $(0,2)$ and $(1,3]$. Do I need to take the different ...
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### Derivative of cross product w.r.t. a vector

How to compute the derivative of $\vec{a}\times\vec{b}$ w.r.t. $\vec{c}$, all of which are 3D vectors for simplicity? Here $\vec{a}(\vec{c})$ and $\vec{b}(\vec{c})$ are both dependent on $\vec{c}$. I ...
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### How to determine the reflex angles in a concave polygon in 3D?

For a concave polygon in 2D, it's easy to use the cross product to determine the reflex angles, which are greater than $180^{\circ}$, but I wonder if there is a simple way to do it in 3D.
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### $\hat i\times\hat j$

Very basic question ahead. It is required to evaluate the cross product of $\hat i$ and $\hat j$, that is, $\hat i\times\hat j$. Knowing that the cross product is anti-commutative, I made sure to ...
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### How to prove the Grassman identity: $(a\times b)\times c=(a\cdot c)b-(a\cdot b) c$ [closed]

I was reading a book and I found this question : how to prove the Grassman identity : (a × b) × c=(a ⋅ c) b−(a ⋅ b) c where a,b and c are vectors. There was a hint that I should begin by expanding the ...
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### Where is the magical sign change under change of basis ? Not pseudotensor?

I'm sorry for the long post but the this subject is confusing to me. Context: On one hand wiki talks about pseudovectors as if they are maps $\Phi:V^k \to V$ on the physical vector space with the ...
1 vote
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### How to determine a vector in $\mathbb R^3$ from its dot and cross products with a given vector

Suppose we have a vector $a$ in $\mathbb R^3$ and an unknown vector $v$, but we know $a \cdot v$ and $a \times v$. Can we find $v$? How? Sources: Based on Shifrin's Multivariable Mathematics and MIT ...
1 vote
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### Question in pseudovectors

I learnt that under parity transformation a vector $\vec{A}$ <---(Parity)------> $-\vec{A}$ and a pseudovector can be written as $\vec{c}=\vec{A}$ $\times$ $\vec{B}$ and since A goes negative A ...
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### Why can the del operator cross product a triple integral be placed inside the triple integral?

Consider the (electric) vector field $$\pmb{E}(\pmb{r})=k_e\iiint_V \frac{\rho(\pmb{r_s})}{\lVert \pmb{r}-\pmb{r_s} \lVert^2}\frac{\pmb{r}-\pmb{r_s}}{\lVert \pmb{r}-\pmb{r_s} \lVert}d\tau\tag{1}$$ ...
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### Dot product symbol for divergence, merely a convenience? [duplicate]

As the title says, is it merely a convenience to write the divergence as a dot product? Is there an intuition on the relationship between the geometric interpretation of the divergence and that of the ...
1 vote
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### How is $\nabla (u\cdot A) =u\cdot \nabla A+ u\times (\nabla \times A)$?

This was used in the answer here, in the derivation of the Lorentz force law from the Lagrangian. $u$ and $A$ are vectors, the velocity of the particle and the spacetime dependent Magnetic field As ...