26-10-2023

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



 
 
 
 
 
 
 
 
 