Change of Variable for a Vector from Rectangular to Cylindrical

[Joshuarr]

Homework Statement

1.21 Express in cylindrical components: (a) the vector from C(3, 2,−7) to
D(−1, −4, 2); (b) a unit vector at D directed toward C; (c) a unit vector at D
directed toward the origin.

I just want to know (a).

And the solution from the book is attached, too.

Homework Equations

ρ = Sqrt(x^2+y^2)
φ = Atan(y/x)
z = z

$$J(r,\phi, z)=\begin{bmatrix} {dx\over dr} & {dx\over d\phi} &{dx\over dz} \\ {dy\over dr} & {dy\over d\phi} & {dy\over dz} \\ {dz\over dr} & {dz\over d\phi} & {dz\over dz}\end{bmatrix} =\begin{bmatrix} {d(r\cos\phi)\over dr} & {d(r\cos\phi)\over d\phi} & {d(r\cos\phi)\over dz} \\ {d(r\sin\phi)\over dr} & {d(r\sin\phi)\over d\phi} & {d(r\sin\phi)\over dz} \\ {dz\over dr} & {dz\over d\phi} & {dz\over dz}\end{bmatrix} =\begin{bmatrix} \cos\phi & -r\sin\phi & 0 \\ \sin\phi & r\cos\phi & 0 \\ 0 & 0 & 1 \end{bmatrix}$$

The Attempt at a Solution

I posted a rather detailed account of what I did, but apparently I took longer than 15 mins or whatever, and it all got erased... I think the physics forum should look into away to prevent that!

Anyway, I tried solving it and I got that the vector from C to D is D-C = (-4,-6,9)

V_cd = -4 a_x - 6 a_y + 9 a_z

So I found ρ = Sqrt(6^2+4^2) = Sqrt(52) = 7.2111
And I found φ = Atan(-6/-4) = 56.3099
And we add 180 to this to get φ = 236.3099 degrees

When I tried finding each component, I got ρ = 7.2111, φ = 0, and z = 9

... In short, I'm very confused. I can't even get the component of ρ of V_cd, which I think should be rather easy.

Also, I'm not even sure how to describe a position vector in cylindrical coordinates.

Is it just r(ρ,φ,z) = ρ a_ρ + φ a_φ + z a_z, where φ is in radians?

In that case, I'd get: V_cd = 7.2111 a_ρ + 4.124 a_φ + 9 a_z

If I do it the other way, where I got each component by taking the dot product of V_cd with each unit vector in cylindrical coordinates (i.e. a_ρ, a_φ, a_z), I would get

V_cd = 7.2111 a_ρ + 9 a_z

And as you may note, neither is like the back of the book. :(

 

[vela]

The two of the unit vectors in cylindrical coordinates aren't constant, so you have a set of unit vectors $\hat{a}_\rho$, $\hat{a}_\phi$, and $\hat{a}_z$ for the point C, and another set for the point D. The vector $\vec{V}_\text{CD}$ has its tail at C and tip at D, so you want to express it in terms of the unit vectors for point C.

What you found was how to write $\vec{V}_\text{CD}$ in terms of the unit vectors for (-4, -6, 9).

 

[Joshuarr]

I'm not sure what you're telling me. If I want the position vectors associated with C and D in terms of constant unit vectors, then I can only represent that in terms of a_x, a_y, and a_z. I can't make a_ρ, a_φ constant, because they're not -- by definition.

So the only way to express the position vectors C and D (with no ambiguity) is in terms of a_x, a_y, a_z... or am I missing something?

I managed to find the vector V_cd in rectangular coordinates. I'm pretty sure that one's correct, and I can find ρ, φ, and z, too. What doesn't make sense to me is how to get it in the form of the bases (the unit vectors). Say, I know ρ, φ, z - what do I do next?

I'm beginning to think the answer in the back of the book is wrong. I've worked on this problem in multiple ways and have not been able to get what they got.

Also, I now know (after reading online) that r(ρ,φ,z) = ρ a_ρ + φ a_φ + z a_z is incorrect because φ is an angle, which is not in units of length.

The ρ component should definitely be ρ = Sqrt(4^2 + 6^2) = Sqrt(52), but they didn't even get that part right.

 

[vela]

I've attached a picture for an example in two dimensions using polar coordinates.

The vector r goes from the origin to some point in the plane. The vector a_r is the unit radial vector. In terms of a_r, you always have r=r a_r, where r=|r|. There's never a component in the a_θ direction because that unit vector is always perpendicular to r.

The vector V, the way I drew it here, is equal to V=a_r+a_θ.

In your problem, V_CD should be in the role of V in my example. (The point C would be the green dot, and the point D would be at the tip of the blue vector.) But you're calculating its components as if it were in the role of r. When you do that, you always get that it's just some multiple of a_r, which is essentially what you're finding (if we neglect the z-component).

 

[Joshuarr]

Wow. Thanks a lot! I finally get it.

So my biggest problem was φ, because this determined a different a_ρ and a_φ.

The value of φ associated with point C is φ_c = Atan(2/3) = 33.69 degrees

Thus,
ρ component of V_cd = (-4 a_x -6 a_y + 9 a_z) dot a_ρ
= -4*cos(φ_c) - 6*sin(φ_c) + 9*0
= -6.6564
φ component of V_cd = (-4 a_x -6 a_y + 9 a_z) dot a_φ
=4*sin(φ_c) - 6*cos(φ_c) + 9*0
=-2.7735
z component of V_cd = z = 9

So the final answer is V_cd = -6.6564 a_ρ - 2.7735 a_φ + 9 a_z.
Just like the book said!

haha. I've spent so many hours on this one problem, but I'm sure it'll help me with future ones.

Thanks for your help again, Vela!

 

[vela]

You're welcome. This is one of those things that makes perfect sense once you get it, but until then, it can be terribly confusing.