i want the proof for unit vector(r)=magnitude of(r) . the directional vector(r0)

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Clarification of Question by guggilla-ga on 14 Apr 2004 11:34 PDT

I want this proof for sperical coorinates

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Request for Question Clarification by hedgie-ga on 16 Apr 2004 14:05 PDT

I am afraid that is quite unclear.

Perhaps you can eleaborate and give some references for context?

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Clarification of Question by guggilla-ga on 19 Apr 2004 09:05 PDT

its given in the book mathematical methods for physicists by arfken
and weber 5th edition page no 123 eq 2.43

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Request for Question Clarification by hedgie-ga on 20 Apr 2004 11:09 PDT

Hi guggilla-ga

I am required to respond to a clarification:

I do not have that particular book and for $2 
I cannot run to the library to get it. 
However, commenters gave  you good and correct answer, 
and it is free :-) so I hoe you are OK with that.

I hope your studies are progressing well.

hedgie

Are you asking to prove that the unit vector in the direction of v is
equal to v divided by the magnitude of v (unit(v)=v/mag(v))? I don't
know what that dot can mean, since magnitude is a scalar, not a
vector.

But if you do mean what I referred to, you can just use the fact that
mag(a*v)=a*mag(v) for any scalar "a" and vector "v". This means that
mag(v/mag(v))=(1/mag(v))*mag(v)=1, so v/mag(v) is a unit vector. Also,
v/mag(v) is a positive scalar multiple of v, so it's in the same
direction. Any unit vector in the same direction as v is v's unique
unit vector. That's a proof. But it might not be of what you want.

From Arfken and Weber 5th ed p123 eq2.43

A position vector r may be written 
R=R'r=R'(x^2+y^2+z^2)^(1/2)
 =X'x+Yy'+Z'z
 =X'rsin(theta)cos(phi) + Y'rsin(theta)sin(phi)+Z'rcos(theta)
where I have substituted ' for the ^ above the vectors i.e. r with a ^
in the book is r' here, and I have substituted capital letters for
bold letters i.e. a bold r in the book is R here.

The first two lines of this do not require spherical coordinates.  

Line 1.  

For any vector R we can express it as a unit vector in that direction
(denoted R') and its length r.   Hence the R=R'r part.

In cartesian coordinates (double application of pythagorean theorem)
r=(x^2+y^2+z^2)^(1/2)
(i presume you know how to derive that since your question is about
spherical coordinates).

Line 2.  
You can also write any position vector R as X+Y+Z where X, Y, Z are
vectors in the x, y, z directions only.  In that case form the 1st
paragrpah about line 1 you can write
X=X'x
Y=Y'y
Z=Z'z
thus line 2

Line 3.
Line 3 is derived from line 2 with the observation that 
x=rsin(theta)cos(phi)
y=rsin(theta)sin(phi)
z=rcos(theta)
which is equations 2.36 from the book.

If you want a proof for eq 2.36 then I can provide that.

Regards