Directions:  Work each problem on clean paper.  Your solutions should be clear, error-free, cogent, and didactic.  You should assume that you are writing so that a C calculus student can understand how to solve the problem by reading your solution.  To that end, writing short explanations and notes are a good.   e.g. “Strategy: Use the fact that the dot product of a vector in the plane and a vector normal to the plane must equal zero to find the equation of the plane.”

1. Neatly draw the region described by .

• Suppose that  and .  Further suppose that .   Use geometry to demonstrate (i.e. find) the vector sum .   Then, find  .

• Carefully plot the point (2, 4, 8) on a 3-dimensional graph.  Your plot should show reasonable visual perspective (i.e. it should look like it really is sitting at the point (2, 4, 8)).

• Carefully draw the line segment  that connects  and .  Include dotted vertical lines from the xy-plane to P and Q to show perspective.

• Find the distance between P and Q, from the previous problem.  Then find the coordinates of the midpoint of the line segment  using the point P and the vector .

• Find a set of parametric equations that represents the line passing through P =  in the direction of the vector .  Since   when , test your parametric equations by verifying the line passes through both  and .

• Find the symmetric equations for the line in the previous problem.

• Carefully draw a 3-dimensional graph of the line from the previous problem.  Show all relevant points.

• Find a set of parametric equations that represents the line passing through P =  and .  Test your parametric equations by verifying the line passes through both points.

1. Find the symmetric equations for the line in the previous problem.

1. Carefully draw a 3-dimensional graph of the line from the previous problem.

1. Line    passes through the points  and .  Line  passes through    and .  Determine if the lines  and  are parallel, intersect, or skew.

1. Find the intersection of line  from the previous problem and the line passing through  that is parallel to .

1. If    and  ,  find the vector projection of u onto v.  (i.e. )

1. If   and  .  Find the vector projection of  onto .

1. Explain what steps were required to derive the vector projection, a 2-dimensional picture might be helpful to this end. 

1. Find the parametric representation of the line passing through the point  parallel to the vector .  Suppose that the vector   is normal (i.e. perpendicular) to the line you just found.  What is the value of s?

1. Suppose that    and  ,  find   such that    and .

1. Explain why the dot product of two vectors must be commutative.

•  Let  ,   ,  and   .  Show that the equation    holds true.

• What is the significance of the previous problem (as opposed to just providing a concrete example)?

• Find the cross product of  and  .

• Show that, in general,    is orthogonal to .