Key Points

The 3D coordinate system consists of three mutually perpendicular lines called the x-axis, y-axis, and z-axis. These axes intersect at a single point called the origin O(0, 0, 0).

The three axes determine three coordinate planes: the XY-plane (equation z=0), the YZ-plane (equation x=0), and the ZX-plane (equation y=0). These planes are mutually perpendicular.

The three coordinate planes divide the space into eight regions called octants. The signs of the coordinates $(x, y, z)$ determine the specific octant in which a point lies.

The coordinates of a point P are represented by an ordered triplet $(x, y, z)$, where x, y, and z are the perpendicular distances from the YZ, ZX, and XY planes, respectively.

Any point on the x-axis has coordinates of the form $(x, 0, 0)$. Similarly, a point on the y-axis is $(0, y, 0)$, and a point on the z-axis is $(0, 0, z)$.

Any point in the XY-plane has coordinates of the form $(x, y, 0)$. A point in the YZ-plane is $(0, y, z)$, and a point in the ZX-plane is $(x, 0, z)$.

For the first octant (I), all coordinates $(+, +, +)$ are positive. For the second octant (II), the coordinates are $(-, +, +)$, and so on for all eight octants.

The distance between two points $P(x_1, y_1, z_1)$ and $Q(x_2, y_2, z_2)$ is given by the formula $PQ = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$.

The distance of any point $P(x, y, z)$ from the origin $O(0, 0, 0)$ is calculated using the formula $OP = \sqrt{x^2 + y^2 + z^2}$.

Three points A, B, and C are collinear if they lie on the same line. This is verified if the sum of the lengths of any two line segments equals the length of the third, for example, $AB + BC = AC$.

Using the distance formula, a triangle is isosceles if two sides are equal, equilateral if all three sides are equal, and right-angled if it satisfies the Pythagoras theorem ($a^2 + b^2 = c^2$).

A quadrilateral ABCD is a parallelogram if its opposite sides are equal in length, that is, $AB = CD$ and $BC = DA$. Additionally, its diagonals AC and BD are not necessarily equal.

To find the equation for a set of points satisfying a geometric condition, assume a general point $P(x, y, z)$ and translate the condition into an algebraic equation involving x, y, and z.