Homework 1
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Physics 3309 Homework 1

Chapter 1

1-3. 
                          
Denote the original axes by , , , and the corresponding unit vectors by ,, . Denote the new axes by , ,  and the corresponding unit vectors by , , . The effect of the rotation is , , . Therefore, the transformation matrix is written as:
                          

1-7. 
                                      
There are 4 diagonals:
, from (0,0,0) to (1,1,1), so (1,1,1) – (0,0,0) = (1,1,1) = ;
, from (1,0,0) to (0,1,1), so (0,1,1) – (1,0,0) = (–1,1,1) = ;
, from (0,0,1) to (1,1,0), so (1,1,0) – (0,0,1) = (1,1,–1) = ; and
, from (0,1,0) to (1,0,1), so (1,0,1) – (0,1,0) = (1,–1,1) = .
The magnitudes of the diagonal vectors are
                             
The angle between any two of these diagonal vectors is, for example,
                          
so that
                                         
Similarly,
                          

1-9.                
a)                   
                       
                          
b) 
                                              
The length of the component of B along A is B cos q.
                                                 
                          
The direction is, of course, along A. A unit vector in the A direction is
                                                    
So the component of B along A is
                                                      
c)                  ;
                                                              
d) 
                                            
e)                    
                      
                          

1-13. Using the Eq. (1.82) in the text, we have
                          
from which

1-19. 
                                               
a) We begin by noting that
                                                      (1)
We can also write that
                           (2)
Thus, comparing (1) and (2), we conclude that
                                                      (3)
b) Using (3), we can find :
                           (4)
so that
                                                      (5)

1-22. To evaluate  we consider the following cases:
a) 
b) 
c) 
d) 
e) 
f) 
g) : This implies that i = k or i = j or m = k.
Then,  for all
h)  for all
Now, consider  and examine it under the same conditions. If this quantity behaves in the same way as the sum above, we have verified the equation
                              
a) 
b) 
c) 
d) 
e) 
f) 
g) 
h) 
Therefore,
                                                        (1)
Using this result we can prove that
                          
First . Then,

Therefore,
                                                      (2)
1-23. Write
                          
                          
Then,

Therefore,

1-25. 
                                     
The unit vectors in spherical coordinates are expressed in terms of rectangular coordinates by
                                       (1)
Thus,

                                                              (2)
Similarly,
                          
                                                              (3)
                                                          (4)
Now, let any position vector be x. Then,
                                                                                             (5)
                                         (6)
                           (7)
or,

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