Projection of a Vector on a Line - Exercises

   Exercise 1   Recall that the projection of a vector      onto the line      through the origin and in the direction of the nonzero vector  y  is

1. Which of the following is true?
   1. This projection is a real number.      |  Answer  |
   2. This projection is a vector.      |  Answer  |
   3. This projection is a line.      |  Comment  |
       
2. If   proj_L(x) 0,   what can you conclude about  x  in relation to the line   ?  |  Answer  |
    

   Exercise 2   Find the following projections

1. Find the projection of  x (4,6)  onto the line in the direction of  y (1,2).       | Answer | Solution  |
    
2. Find the projection of   x (1,1)   onto the line   ,    3,  where    is in  .        |  Answer  |  Solution  |
    
3. Find the projection of   x (2,3)   onto the line  2 0.
    |  Answer  |  Solution  |

   Exercise 3  

1. Find the projection of   x (2,1,2)   onto the line  x a, where  a (1,1,3)  and    is in  .    |  Answer  |  Solution  |
2. Find the projection of   x (1,1,4)   onto the line
    x 3,  2,  4,   where    is in  .
            |  Answer  |  Solution  |

   Exercise 4 (orthogonalizing formula)   Given an nonzero vector  a  and an arbitrary vector  b, show that the vector

is perpendicular to  a. - This orthogonalizing formula is the key ingredient in the orthonormalization algorithm of Gram and Schmidt.    |  Hint  |    |

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