3
votes
1answer
109 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 ...
0
votes
0answers
10 views

Cross Product of Covectors

Is the vector/cross product defined for covectors (in the dual space) or is it, strictly speaking, only defined for vectors themselves? I would imagine that it works fine for covectors but I wanted to ...
3
votes
1answer
61 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?
2
votes
3answers
470 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
70 views

Possible to solve this coupled system of vector equations?

Let $\gamma, \omega, c$ be positive constants, let $\mathbf{Q}_{a}$ and $\mathbf{Q}_{b}$ be three-dimensional vectors, and let $\mathbf{B}(\mathbf{r})=\mathbf{B}(x,y,z)$ be a vector field. Let ...
5
votes
1answer
561 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 ...
1
vote
1answer
185 views

proof for $[\vec{a}\cdot (\vec{b} \times \vec{c})]\vec{a}=(\vec{a}\times\vec{b})\times(\vec{a}\times\vec{c})$

I encounter this triple product property in wikipedia But I can't find proof for $$[\vec{a}\cdot (\vec{b} \times \vec{c})]\vec{a}=(\vec{a}\times\vec{b})\times(\vec{a}\times\vec{c})$$ The RHS cross ...
0
votes
1answer
56 views

Derivative of a vector function

Can someone please check my work below to confirm whether or not I got the correct answer? This is question 13.2.16 in the 7th edition of Stewart Calculus. Find the derivative of the vector function: ...
0
votes
1answer
43 views

problem understanding $a\times((b\cdot b)a-(b\cdot a)b)=-a\cdot ba\times b$

$a\times(b\times(a\times b))=a\times((b\cdot b)a-(b\cdot a)b)=-a\cdot ba\times b$ can anyone expand on how the final answer is derived? I try to expand but ended up scratching my head. the best I can ...
2
votes
2answers
228 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 ...
1
vote
1answer
6k views

Fleming's “right-hand rule” and cross-product of two vectors

I have been throwing around hand gestures for the past hour in a feeble attempt at trying to solve this question involving a cross product of two vectors $a$ x $b$. So far, I haven't found any ...
0
votes
1answer
1k views

normalized cross product

Is there a way to get the result of a cross product to be normalized after just a cross action, i.e. without doing after the cross v/|v|? (the vectors involved are ...