Flash Cards: Introduction to Three Dimensional Geometry

In three-dimensional space, a point is defined by a triplet of real numbers (x, y, z), which represent its distances from three mutually perpendicular coordinate planes: the XY-plane, YZ-plane, and ZX-plane.

The coordinate axes in 3D geometry are the x-axis, y-axis, and z-axis. They are mutually perpendicular lines that intersect at the origin (0, 0, 0).

The origin in 3D space is denoted by the coordinate (0, 0, 0). It is the point where all three coordinate axes intersect.

To find the coordinates of a point P in space, drop a perpendicular to the XY-plane to get y, then to the x-axis to get x, and finally measure height z from the XY-plane.

The XY-plane is the flat surface formed by the x-axis and y-axis, where the z-coordinate is zero (z = 0). All points in this plane have coordinates (x, y, 0).

There are eight octants in 3D space, categorized based on the signs of the coordinates (positive or negative) of points within them.

The coordinates of any point on the x-axis are of the form (x, 0, 0), where y and z are both zero.

The coordinate x indicates the distance from the YZ-plane, y from the ZX-plane, and z from the XY-plane. Their signs determine the octant.

Points in the YZ-plane have coordinates of the form (0, y, z), where x is zero.

To plot (x, y, z), first plot (x, y) on the XY-plane, then from there, measure up or down to locate z.

Distances above the reference planes are positive, while distances below are negative. For example, a height above the XY-plane is positive.

The XY-plane has z = 0, while the YZ-plane has x = 0. Points on the XY-plane contain only x and y coordinates.

The triplet (x, y, z) uniquely defines the location of a point in three-dimensional space, allowing for precise geometric interpretations.

Perpendicular distances are the shortest distances from a point to a plane, calculated along the line that is normal (perpendicular) to that plane.

A common mistake is mixing up the coordinates, such as confusing x with y or misjudging positive and negative directions along axes.

To determine the octant of (x, y, z), check the signs of x, y, and z: positive values for octant I, negative for octant III, etc.