Math 2374 Review of some material about vectors.

One section of the roof of a house has a rectangular hole in it. In some coordinate system, the vertices of this rectangular hole are (4,5,6), (5,7,9), (5,3,7) and (6,5,10).

1. Prove that this hole really has a rectangular shape.

2. Find the area of the hole. (Assume that all units are in inches.

3. This section of the roof forms part of a plane. Show that the equation of this plane is 4x+y-2z = 9

4. Find a vector n which is normal to this plane. Choose a normal which points down into the house.

It is raining at the rate of 2 inches per hour. Because of the wind, the rain does not fall straight down, but rather in the direction of the vector v = (1,4,-9).

5. Find the projection of v onto n. (In Math 1372, this was called the vector projection)

6. Find the component of v in the direction of n. (In Math 1372, this was called the scalar projection.)

7. At what rate is the rainwater flowing through the window? Your answer should be in cubic inches per hour.