The document provides solutions to problems finding the direction cosines and direction angles of various vectors in 3-space. In general, it is shown that the direction cosines and angles can be found by taking the ratio of the vector components to the vector's magnitude and using trigonometric functions. Several examples are worked through, finding the direction cosines and expressing the direction angles in radians.

1. 1
(MMP) Problem No.10
Find the direction cosines
and direction angles of the
vector 2 5 4
  
a i j k
SOLUTION
1 2 3
1
3
2
2 2 2
We know that for a nonzero
vector
in 3-space, the angles , ,
& between and the unit
vectors , ,& ,respectively,
are given as:cos ,
cos , &cos
such that
cos cos cos 1
a a a
a
a
a
 


 
  
  

 
  
a i j k,
a
i j k
a
a a

2. 2
2 5 4
4 25 16 45 3 5
2 5 5
cos ,cos ,
3
3 5 3 5
4
cos
3 5
4 5 16 4 25 16
1
45 9 45 45
 

  
     
   

 
    
a i j k
a
1 1
0
1 1
0
1 1
0
,
2
cos ( ) cos (0.298)
3 5
1.268 72.662
5
cos ( ) cos (0.745)
3
0.73 41.84
4
cos ( ) cos (0.596)
3 5
0.932 53.416
Now
rad
rad
rad



 
 
 
 
 
 
 
 
 
(MMP) Problem No.11

3. 3
Find the direction cosines
and direction angles of the
vector 2 3
  
a i j k
SOLUTION
1 2 3
1
3
2
2 2 2
We know that for a nonzero
vector
in 3-space, the angles , ,
& between and the unit
vectors , ,& , respectively,
are given as:cos ,
cos , &cos
such that
cos cos cos 1
a a a
a
a
a
 


 
  
  

 
  
a i j k,
a
i j k
a
a a

4. 4
2 3
1 4 9 14
1 2
cos ,cos ,
14 14
3
cos
14
1 4 9
1
14 14 14
 

  
    
  

   
a i j k
a
1 1
0
1 1
0
1 1
0
,
1
cos ( ) cos (0.267)
14
1.30 74.51
2
cos ( ) cos (0.534)
14
1.01 57.72
3
cos ( ) cos (0.802)
14
0.64 36.678
Now
rad
rad
rad



 
 
 
 
 
 
 
 
 
(MMP) Problem No.12

5. 5
Find the direction cosines
and direction angles of the
vector 6 6 3
 
a i j - k
SOLUTION
1 2 3
1
3
2
2 2 2
We know that for a nonzero
vector
in 3-space, the angles , ,
& between and the unit
vectors , , & , respectively,
are given as:cos ,
cos , & cos
such that
cos cos cos 1
a a a
a
a
a
 


 
  
  

 
  
a i j k,
a
i j k
a
a a
6 6 3
36 36 9 81 9
6 2 2
cos ,cos ,
9 3 3
3 1
cos
9 3
4 4 1
1
9 9 9
 

 
     
   

  
   
a i j - k
a

6. 6
1
1
0
1 1
0
,
2
cos ( )
3
cos (0.666)
0.842 48.241
1
cos ( ) cos ( 0.333)
3
1.91 109.45
Now
rad
rad
 



 
 

 

  
 
(MMP) Problem No.13
Find the direction cosines
and direction angles of the
vector 3

a i - k
SOLUTION
1 2 3
1
3
2
2 2 2
We know that for a nonzero
vector
in 3-space, the angles , ,
& between and the unit
vectors , , & , respectively,
are given as:cos ,
cos , & cos
such that
cos cos cos 1
a a a
a
a
a
 


 
  
  

 
  
a i j k,
a
i j k
a
a a

7. 7
2 2
3
1 3 4 2
1
cos 0.50,
2
cos 0,
3
cos 0.866
2
1 3
(0.50) ( 0.866)
4 4
0.25 0.75 1




    
  

   
    
  
a i - k
a

8. 8
1 1
0
0
1 0
0
1 1
0
0
,
1
cos ( ) cos (0.50)
2
1.05 60
3.14
(60 1.047)
3 3
cos (0) 1.571 90
3.14
(90 1.571)
2 2
3
cos ( ) cos ( 0.866)
2
2.619 150
5 15.71
(150 2.619)
6 6
Now
rad
rad
rad






 

 
 
 
  
  
  
   
 
  
(MMP) Problem No.14
Find the direction cosines
and direction angles of the
vector 5 7 2
  
a i j k
SOLUTION

9. 9
1 2 3
1
3
2
2 2 2
We know that for a nonzero
vector
in 3-space, the angles , ,
& between and the unit
vectors , , & , respectively,
are given as:cos ,
cos , & cos
such that
cos cos cos 1
a a a
a
a
a
 


 
  
  

 
  
a i j k,
a
i j k
a
a a
5 7 2
25 49 4 78
5 7
cos ,cos ,
78 78
2
cos
78
25 49 4
1
78 78 78
 

  
    
  

   
a i j k
a

10. 10
1 1
0
1 1
0
1 1
0
,
5
cos ( ) cos (0.566)
78
0.969 55.528
7
cos ( ) cos (0.793)
78
0.655 37.53
2
cos ( ) cos (0.227)
78
1.34 76.88
Now
rad
rad
rad



 
 
 
 
 
 
 
 
 