01 3-1 (a) The three-dimensional orthogonal coordinate system
Why three axes
- On a line, a body's position is one number: its distance from the origin .
- In a plane, two perpendicular axes give the pair by projecting the point onto each axis.
- In space we need three mutually perpendicular axes , and meeting at the origin .
- A point is given by the ordered triple , the projections of on the three axes. The set of all such triples is .
The right-hand rule
- Curl the fingers of the right hand from the positive -axis towards the positive -axis; the thumb then points along the positive -axis.
- Every system in this course follows this rule (a right-handed system).
The coordinate planes
- All points lie in the -plane, whose equation is .
- All points lie in the -plane, whose equation is .
- All points lie in the -plane, whose equation is .
- The three planes divide space into eight regions called octants.
Plotting a point
✍️ Worked examples
Example 1: Locate and .
Solution:
- For : plot in the -plane, then move 2 units in the positive direction.
- For : the -coordinate is , so lies in the -plane. Take 4 on the -axis, then move 1 unit in the negative direction.
Distances to planes and axes
- Distance from to the -plane is ; to the -plane it is ; to the -plane it is .
- Distance to the -axis is ; to the -axis it is ; to the -axis it is .
- For example, the distance from to the -axis is .
- A point on the -axis has the form ; on the -axis ; on the -axis .