MTH113 Math & Art: Linear PerspectiveThe Three-Dimensional Cartesian Coordinate System

1. As mentioned in the notes, all points in the xy-coordinate plane have z-value equal to 0. In fact, z = 0is the equation of the xy-plane.

a) What is the equation of the xz-coordinate plane?

b) What is the equation of the yz-coordinate plane?

2. In each of parts a), b), and c), we consider a rectangular box with sides parallel to the coordinateplanes. Some of the coordinates of the eight corners of the box are given; find the missing coordinates.

a) (1, 1, 1), (1, 1, 5), (4, 1, 5), (4, 1, 1), (4, 7, 1), (4, 7, 5), (1, 7, 5), (?, ?, ?)

b) (1, 2, 3), (2, 3, 4), (?, ?, ?), (?, ?, ?), (?, ?, ?), (?, ?, ?), (?, ?, ?), (?, ?, ?)

c) (1, 1, 1), (3, 4, 5), (?, ?, ?), (?, ?, ?), (?, ?, ?), (?, ?, ?), (?, ?, ?), (?, ?, ?)

d) Which of the above boxes is a cube? How big is it?

3. As a prelude to the Perspective Theorem, suppose the art director was at a viewing distance, d = 3,from the window (xy-plane); and suppose that the point P (x, y, z) in the real world had coordinatesx = 0, y = 4, z = 5.

a) Which coordinate plane would P (x, y, z) lie in?

b) What would x of its projection P (x, y, z) onto the window be? (Easy!)

c) What would y be?

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