Hi i know this is a really really simple question but it has me confus

rastafarral6 2021-11-13 Answered
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
\(\displaystyle\vec{{{a}}}\times\vec{{{r}}}\)
The vectors are given by
\(\displaystyle\vec{{{a}}}={a}\vec{{{z}}},\ \vec{{{r}}}={x}\vec{{{x}}}+{y}\vec{{{y}}}+{z}\vec{{{z}}}\)
The vector \(\displaystyle\vec{{{r}}}\) is the radius vector in cartesian coordinares.
I want to calculate the cross product in cylindrical coordinates, so I need to write \(\displaystyle\vec{{{r}}}\) in this coordinate system.
The cross product in cartesian coordinates is
\(\displaystyle\vec{{{a}}}\times\vec{{{r}}}=-{a}{y}\vec{{{x}}}+{a}{x}\vec{{{y}}}\),
however how can we do this in cylindrical coordinates?

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Plainmath recommends

  • Ask your own question for free.
  • Get a detailed answer even on the hardest topics.
  • Ask an expert for a step-by-step guidance to learn to do it yourself.
Ask Question

Expert Answer

Donald Proulx
Answered 2021-11-14 Author has 1699 answers
The radius vector \(\displaystyle\vec{{{r}}}\) in cylindrical coordinates is \(\displaystyle\vec{{{r}}}={p}\vec{{{p}}}+{z}\vec{{{z}}}\). Calculating the cross-product is then just a matter of vector algebra:
\(\displaystyle\vec{{{a}}}\times\vec{{{r}}}={a}\vec{{{z}}}\times{\left({p}\vec{{{p}}}+{z}\vec{{{z}}}\right)}\)
\(\displaystyle={a}{\left({p}{\left(\vec{{{z}}}\times\vec{{{p}}}\right)}+{z}{\left(\vec{{{z}}}\times\vec{{{z}}}\right)}\right)}\)
\(\displaystyle={a}{p}{\left(\vec{{{z}}}\times\vec{{{p}}}\right)}\)
\(\displaystyle={a}{p}\vec{{\phi}}\),
where in the last line we've used the orthonormality of the triad \(\displaystyle{\left\lbrace\vec{{{p}}},\vec{{\phi}},\vec{{{z}}}\right\rbrace}\)
Have a similar question?
Ask An Expert
0
 

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Relevant Questions

asked 2021-04-07
Ok, If a bobsled makes a run down an ice track starting at 150m vertical distance up the hill and there is no friction, what isthe velocity at the bottom of the hill?
I know that the initial velocity here is 0 because it isstarting from rest. And this problem deal with theconservation of energy. But, I don't know where to go fromhere.
asked 2021-09-26
Find the unit vector that has the same direction as the vector v.
\(\displaystyle{v}=-{i}+{2}{j}\)
asked 2021-08-09

This question has to do with binary star systems, where 'i' is the angle of inclination of the system.
Calculate the mean expectation value of the factor \(\displaystyle{{\sin}^{{{3}}}}\) i, i.e., the mean value it would have among an ensemble of binaries with random inclinations. Find the masses of the two stars, if \(\displaystyle{{\sin}^{{{3}}}}\) i has its mean value.
Hint: In spherical coordinates, \(\displaystyle{\left(\theta,\phi\right)}\), integrate over the solid angle of a sphere where the observer is in the direction of the z-axis, with each solid angle element weighted by \(\sin^3 \theta\).
\(\displaystyle{v}_{{{1}}}={100}{k}\frac{{m}}{{s}}\)
\(\displaystyle{v}_{{{2}}}={200}{k}\frac{{m}}{{s}}\)
Orbital period \(\displaystyle={2}\) days
\(\displaystyle{M}_{{{1}}}={5.74}{e}^{33}{g}\)
\(\displaystyle{M}_{{{2}}}={2.87}{e}^{33}{g}\)

asked 2021-05-05
If John, Trey, and Miles want to know how’ | many two-letter secret codes there are that don't have a repeated letter. For example, they want to : count BA and AB, but they don't want to count“ doubles such as ZZ or XX. Jobn says there are 26 + 25 because you don’t want to use the same letter twice; that’s why the second number is 25.
‘Trey says he thinks it should be times, not plus: 26-25, Miles says the number is 26-26 ~ 26 because you need to take away the double letters. Discuss the boys’ ideas, Which answers are correct, which are not, and why? Explain your answers clearly and thoroughly, drawing ‘on this section’s definition of multiptication.. -
asked 2021-02-21
How do you solve this problem? I don' t even know whereto begin.
A Ferrari with a mass of 1400 kg approaches a freeway underpassthat is 10 m across. At what speed must the car be moving, inorder for it to have a wavelength such that it might somehow"diffract" after passing through this "single slit"? How dothese conditions compare to normal freeway speeds of 30m/s?
asked 2021-11-13
I'm having trouble with these types of questions. I have the following vector \(\displaystyle{u}={\left({4},{7},-{9}\right)}\) and it wants me to find 2 vectors that are perpendicular to this one.
I know that \(\displaystyle{\left({4},{7},-{9}\right)}\cdot{\left({x},{y},{z}\right)}={0}\)
asked 2021-09-19
Find the scalar and vector projections of b onto a.
\(\displaystyle{a}={i}+{j}+{k},{b}={i}-{j}+{k}\)
...