1. Triangle ABC has vertices A(3, 3, 0), B(2, 5, 0), and C(4, 0, 2).

(a) Use the dot product to determine the measure of two angles within the triangle, and then triangle logic to calculate the third angle to the nearest degree.

(b) Based on your findings, and without using the cross product, calculate the area of Triangle ABC to the nearest hundredth of a square unit (You may use any method to find the area except the cross product, and it may be a method that does not use vectors at all. Please be sure to justify your answer).

(c) Using the cross product, calculate the area of the triangle to the nearest hundredth of a square unit .

2) A traffic sign is hanging from two cables that are attached to the bottom of an overpass at angles of 36 degrees and 40 degrees to the horizontal. Use vectors to determine the tension in each cable. You will need to know that the weight of the sign is 1000 Newtons. Express answers to the nearest tenth of a unit. Please include a fully-labeled vector diagram as part of your solution.

3) (a) Use a specific example to prove that the cross product is not associative. That is, use three specific vectors, of your own choosing, in 3-space to show that a ( b c) is not equal to (a b) c.

(b) Prove that for any vector in 3-space, the dot product is distributive. That is, prove that a ( b + c) = a b + a c. [Hint: Let a = (a1, a2, a3) and let b = (b1, b2, b3)].