Terminology and notation

In the $x$-$y$ plane, the coordinate axes break the plane into four quadrants, and the first quadrant is the region where $x \ge 0$ and $y \ge 0$.

In 3-dimensional space, the coordinate planes break space into eight regions, called octants. The first octant is the region where $x \ge 0$, $y \ge 0$ and $z \ge 0$. Unlike in the plane, there is no standard numbering for the other octants.

We usually think of the $x$-$y$ plane as being horizontal, with the $x$ axis pointing East, the $y$ axis pointing North, and the $z$ axis pointing straight up. This is described by the right hand rule. If you hold your right hand so that the fingers are pointing in the positive $x$ direction, and if you can bend your fingers so that they point in the positive $y$ direction, then your outstretched thumb is pointing in the positive $z$ direction.  (How would things be different if we applied the left hand rule instead? Try it and see.)

Points are written as triples of numbers between parentheses, sometimes preceded by the name of the point. So $P(3,2,1)$ is a point named $P$ at $x=3$, $y=2$, $z=1$, while $(3,2,1)$ is an unnamed point at the same spot.

Angle brackets are used to denote vectors, as explained in the next learning module. The displacement from $P(3,2,1)$ to $Q(5,6,7)$ is the vector ${\overrightarrow{PQ}} = \langle 2,4,6 \rangle$.