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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
[tex]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}[/tex]
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 DC = (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. :(
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