12.1 Three Dimensional Coordinate System
Introducing Three Dimensions!
Remember ? We're great at working in the -"plane". But what about ? We could think of this as working in an -"space".
We're used to working with functions with one input and only one output. Typically, we think of our -value as our input (-axis), and our -value as our output (-axis).
We're entering into multi-variable calculus. Multi-variable can mean lots of things, which we'll get into later, but first we should develop a good grasp on the three-dimensional coordinate system. Enter:
Three variables can be thought of as a three dimensional space, just like can be thought of as a plane. Where each of the coordinates tell us how far to move in the direction parallel to the respective axis. Which brings us to the right hand rule, and an explanation of the axes. We will follow the generally accepted convention of thinking of the -axis as being vertical, with the - and -axes being parallel to the ground. You might be wondering, how do we determine the orientation of the - and -axes? Good question: the right hand rule. (us left handed shmucks are out of luck). Draw an imaginary set of axes directly under your right hand with its thumb up. Then think of wrapping your finders around a vertical -axis, keeping your thumb up. Your fingers will sweep across first the positive -axis, and then -axis in a counter clockwise rotation. (<----math's favorite direction! You may need to rotate the entire set of axes to align the positive - and -axes to directly in front of your view point)