Introduction to Three Dimensional GeometryCoordinate Axes and Coordinate Planes in Three Dimensional Space

Look at a corner of your classroom, where the floor meets two walls. Three flat surfaces meet at a single point, and each pair of them meets along a straight edge: two edges run along the floor and the third runs straight up. All three edges are at right angles to one another.

Mathematics takes that corner and extends it forever in every direction. Take three planes through a point O, each one perpendicular to the other two. Each pair of planes cuts along a line, and the three lines X′OX, Y′OY and Z′OZ are called the x-axis, the y-axis and the z-axis. Because the planes are mutually perpendicular, so are the axes. Together they form a rectangular coordinate system, and O is its origin. In every picture in this chapter the x-axis is red, the y-axis green and the z-axis blue.

Each plane is named after the two axes it contains: the plane through the x- and y-axes is the XY-plane, and in the same way we have the YZ-plane and the ZX-plane. These are the three coordinate planes. We usually picture the XY-plane as horizontal — the floor, or the page of your book lying flat on the desk — and then the z-axis stands vertical.

Fig. 11.1 is live. Name the three coordinate planes: tap a plane in the picture, then pick its name. Drag sideways to turn the picture.

Once named, each plane took the colour of the one axis it does not contain — you will see why in the next section. Distances along the axes carry signs, just as on a number line. Measured from the XY-plane, points above it (towards Z) have a positive z and points below it (towards Z′) a negative z. In the same way, y is positive to the right of the ZX-plane (towards Y) and negative to its left, and x is positive in front of the YZ-plane (towards X) and negative behind it.

Eight Rooms Around the Origin

One wall splits a hall into two sides. Three coordinate planes, each cutting through the others, split the whole of space into separate regions. Below, every region is drawn as a block, pulled a little away from the planes so that you can see where they run. Tap a block to read the signs of x, y and z inside it, and complete the four missions. Some blocks hide at the back — turn the picture to reach them.

Count the blocks: the three coordinate planes divide space into eightsixfour parts, called octants. The book names them by the half-axes around them — XOYZ, X′OYZ, X′OY′Z, XOY′Z, XOYZ′, X′OYZ′, X′OY′Z′ and XOY′Z′ — and numbers them I, II, III, …, VIII in that order. Octant I, where all three coordinates are positive, is the three-dimensional cousin of the first quadrant.