How Do I Use Unit Directional Vectors in Vector Calculus?

[jmckennon]

Homework Statement

Attached

Homework Equations

The Attempt at a Solution

I'm not quite sure how to start these. Once I figure out one of them, I should be able to do the rest. The confusing piece to me is with the unit directional vector (a_rho etc. My teacher has me horribly confused as to what I'm supposed to do with those terms. My initial idea is to use the matrix method for solving it. But I'm not sure. Any help would be greatly appreciated!

 

[rarkup]

You might do better if you post your problem as text rather than an attachment - some folks are wary of them.

 

[Saladsamurai]

The unit vectors are just like regular vectors. You simply multiply the scalars which gives you the magnitude and then dot/cross the unit vectors (whichever applies).

I believe that a_rho, a_phi, z are analogous to $\mathbf{i},\mathbf{j},\mathbf{k}$, but don't quote me on that.

If you cross them from 'left to right' (i.e. i x j , j x k, x i) you get the positive 'next one' (i.e. k, i, j , respecively); if you cross them 'right to left' you get the negative 'next one.'

Example:

$$5\mathbf{i}\times6\mathbf{j}=((5)(6))(\mathbf{i}\times\mathbf{j})=30\mathbf{k}$$

And

$$3\mathbf{k}\times2\mathbf{j}=((3)(2))(\mathbf{k}\times\mathbf{j})=6(-\mathbf{i})=-6\mathbf{i}$$Assuming that $a_\rho,a_\phi,z$ form a right-handed coordinate system, you can simply replace:

$a_\rho=\mathbf{i}$
$a_\phi=\mathbf{j}$
$z=\mathbf{k}$

in the preceding examples.

 

[rarkup]

Is your co-ordinate system a Cartesian one - ie are the 3 axes at 90 degrees to each other, as in the usual convention of x, y and z axes?

The reason I ask is because the subscripts in your question, namely rho and phi, are more commonly used with curvilinear co-ordinate systems (these are co-ordinate systems that change as you travel along the curve under investigation).