Math 3118, Section 1

Fall 2002

Class exercises for Wednesday, November 13   

1. Calculating the dot product
   • If  P = (a,b) and  Q = (u,v), then the dot product P·Q is given by the formula:
           P·Q = au + bv.
     Sample: If P = (1,3),  and  Q = (5,-3),  then  P·Q = 1·5 + 3·(-3) = 5 - 9 = -4.
     So, the dot product of two vectors is just a number.
      
   • Calculate  P·Q in the following cases:
     1. P= (2,-1)  and  Q= (4,3)
        Sketch these two points in the plane.
         
     2. P= (2,4)  and  Q= (6,-3).
        Sketch these two points in the plane.
        What seems to be the relationship of the line joining  (0,0)  to  P with the line joining  (0,0)  to Q?
         
     3. P= (4,3)  and  Q = P= (4,3).
        Also, calculate  ||P|| and compare this with the dot product  P·P.
         
     4. P= (a,b)  and  Q = P= (a,b).
        Also, calculate  ||P|| and compare this with the dot product  P·P.
          
2. = Exercise 8.4.2 in the text
   Actually, just look at this one just long enough to see which parts of exercise 1. above take care of
   the "illustrate" and "prove" parts.
    
3. A few more dot product calculations
   
   • Calculate  P·Q  and  Q·P,   and compare the two answers in the following cases:
     1. P= (4,3)  and  Q= (5,-1)
         
     2. P= (a,b)  and  Q= (c,d)
         
   • Write the general equation that gives the relation between  P·Q  and  Q·P.  
      
4. = Exercise 8.4.4 in the text
   Suggestion: You just need to figure out how to modify the conclusion of "Theorem 12", stated in the text,
   immediately above where exercise 8.4.4 is presented. Actually, if you can do that, then it's not any harder
   to see how to modify the calculation in the proof of the theorem in order to justify your conclusion.
   The modification involves using  (P+Q)·(P+Q)  as another way to represent  ||P+Q||².
   Then you use the fact that the dot product "distributes over addition", from exercise 8.4.1;
   the result from exercise 3. above also may be useful . . .
    
5. = Exercise 8.4.7 in the text
   Suggestion: This involves setting up a system of 2 equations in 2 unknowns to find the point  F = (x,y)  
   where 2 of the 3 altitudes of the triangle intersect. Next, solve the equations to find the point. The specifics
   of how to do this aren't fully presented in the text, but the discussion on the bottom half of page 174 does
   at least give a good indication as to what has to be implemented.
       After finding the point, then do an appropriate calculation to verify that the point  F you found also lies
   on the third altitude  --  you may or may not want to do this by the same method that's indicated in the text . . .
    

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