1. Let v = < 0, sqrt(3)/2, 1/2> and complete the following, showing all your work:
 

v dot i =                                                           v cross i =

v dot j =                                                           v cross j =

v dot k =                                                         v cross k =
 
 
 
 

and plot i, j, k, v, v cross i, v cross j, and v cross k below:

 

2. For the following vectors in 1., circle whether or not the statement is T (true) or F (false), and state the reason why:

      statement                                    why                                        statement                                why

   v orth i         T     F                                                                 v | | i          T     F
 
 

   v orth j         T     F                                                                 v | | j         T     F
 
 

   v orth k       T     F                                                                  v | | k        T     F
 

      statement                                       why                                statement                                    why

 (v cross i)  orth i         T     F                                                    (v cross i) | | i          T     F
 
 

  (v cross j)  orth i        T     F                                                    (v cross j) | | i         T     F
 
 

 (v cross k)  orth i        T     F                                                    (v cross k) | | i        T     F
 

3. Find three different nonzero vectors orthogonal to both of the vectors i and v in 1.
 
 
 
 
 
 
 
 
 
 

4. Let a, b, c be three nonzero vectors and prove the following: If  a dot (b cross c) = 0, then either b and c are parallel, or a is in the plane determined by b and c. Use the following cases and draw a diagram for each case.
 

Case 1. (b cross c) = 0
 
 
 
 
 
 
 
 
 
 

Case 2. (b cross c) != 0