
Working with the dot product and cross product.
Work in groups of 3–4, writing your answers on a separate sheet of paper.
Let be a random (nonzero) vector in . Could there be a vector such that one
expects Explain your reasoning.
Consider three vectors in : We claim: Give an algebraic verification of
this claim. For your information, is commonly called the scalar triple
product.
Use the diagram
and recall that the volume of a parallelpiped is given by to answer the
following:
 First, explain what and mean geometrically and how they must be equal. There is a basic reason regarding computing volumes at play here, so you should not appeal to Problem 6 in this part).
 Second, explain why it then follows that is the volume of the parallelepiped spanned by vectors , , and . You may use Problem 6 on this part.