The document finds the angle between two vectors representing the face diagonal and body diagonal of a rectangular block. The vectors are defined as A=ai+aj and B=ai+aj+ak. The angle between the vectors is calculated to be 53.3 degrees, or approximately 70 radians. The document also finds the value(s) of q that make another two vectors, 3=q+q2 and B=i+j+k, perpendicular. Setting their dot product equal to 0 and solving yields that q=0 or q=1 are the values where the vectors are perpendicular.

1. 1
Classical Mechanics P. No.3
Find the angle between the
vectors
2
(Note: These two vectors
define a face diagonal and a
body diagonal of a
rectangular block of sides
a, 2a
a a
and
= a +2a +3a
 
A i j
B i j k
, and 3a.)
Solution
2 2
2
2 2 2
2
2
4
5 5
4 9
14 14
a a
a a
a a
= a + 2a +3a
a a a
a a
 
  
 
   
 
A i j
A
B i j k
B

2. 2
2 2 2
2
2
2
1
,
. 4 5
Let be the angle between
& , then
. 5
cos =
( )( ) 5 14
5 5
0.5976
70 70
cos (0.5976)
0.93 53.3
Now
a a a
a
a Xa
a
a
radian


 
  

  
 
 
A B
A B
A B
A B
Classical Mechanics P. No.4
For what value (or values)
of q is the vector
3
perpendicular to the vector
2 ?
q
q q
  
  
A i j k
B i j k
Solution
2 2
( 3 ( 2 )
3 2 2 2
( 1) 2( 1) ( 1)( 2)
q ) q q
q q q q q
q q q q q
    
      
      
A.B i j k . i j k

3. 3
We know that dot product
(or scalar product) of two
perpendicular vectors is
always zero. Hence, if &
are perpendicular, then
. must be equal to zero.
. 0
( 1)( 2) 0
either 1 0, or 2 0
eit
=
q q
q q

   
    

A
B
A B
A B
her 1 2
q or q
 
Analytical Mechanics, Grant R. Fowles, George L. Cassiday