Introduction to Three-dimensional Geometry

The One Thing

Three-dimensional geometry represents each point uniquely as an ordered triple $(x,y,z)$ measured along three mutually perpendicular axes. Coordinate-wise formulas then determine distances, midpoints, and points dividing a line segment internally or externally.

Definitions and Results

• Three-dimensional coordinate system: A system formed by three mutually perpendicular axes—the $x$-axis, $y$-axis, and $z$-axis—which meet at a common origin and provide a right-handed three-dimensional reference framework.

• Origin: The common point of the three coordinate axes, denoted by $O(0,0,0)$.

• Coordinate axes: The mutually perpendicular $x$-, $y$-, and $z$-axes used as reference lines for measuring the coordinates of points in space.

• Coordinates of a point: A point $P$ is represented uniquely by $P(x,y,z)$, where $x$, $y$, and $z$ are real numbers and are the directed distances from the $yz$-plane, $zx$-plane, and $xy$-plane, respectively.

• Coordinate planes: The planes formed by pairs of coordinate axes. They are:

- $xy$-plane: Contains the $x$- and $y$-axes; its equation is $z=0$. - $yz$-plane: Contains the $y$- and $z$-axes; its equation is $x=0$. - $zx$-plane: Contains the $z$- and $x$-axes; its equation is $y=0$.

• Coordinate axes in coordinate form:

- A point on the $x$-axis has coordinates $(x,0,0)$. - A point on the $y$-axis has coordinates $(0,y,0)$. - A point on the $z$-axis has coordinates $(0,0,z)$.

• Octants: The eight regions into which the three coordinate planes divide space. They are identified by the signs of $x$, $y$, and $z$. In particular, the first octant satisfies

$$x>0,y>0,z>0.$$ A zero coordinate indicates that the point lies on a coordinate plane or, if two coordinates are zero, on a coordinate axis.

• Distance from coordinate planes: For $P(x,y,z)$, the distances are:

$$\text{from the }yz\text{-plane}=|x|,$$ $$\text{from the }zx\text{-plane}=|y|,$$ $$\text{from the }xy\text{-plane}=|z|.$$

• Distance between two points: If $A(x_1,y_1,z_1)$ and $B(x_2,y_2,z_2)$, then

$$AB=\sqrt{(x_2-x_1{)}^2+(y_2-y_1{)}^2+(z_2-z_1{)}^2}.$$

• Midpoint: The midpoint of $A(x_1,y_1,z_1)$ and $B(x_2,y_2,z_2)$ is

$$\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2},\frac{z_1+z_2}{2}\right).$$

• Section formula: The section formula gives the coordinates of a point dividing the line segment joining two given points internally or externally in a specified ratio.

• Internal division: A point lies between the two endpoints and divides the line segment in a positive ratio. If $P$ divides $A(x_1,y_1,z_1)$ and $B(x_2,y_2,z_2)$ internally in the ratio $m:n$, then

$$P=\left(\frac{m{x}_2+n{x}_1}{m+n},\frac{m{y}_2+n{y}_1}{m+n},\frac{m{z}_2+n{z}_1}{m+n}\right).$$ Each coordinate is a weighted average of the corresponding endpoint coordinates.

• External division: A point lies outside the line segment and divides the line joining the endpoints in the specified ratio. If $P$ divides $A(x_1,y_1,z_1)$ and $B(x_2,y_2,z_2)$ externally in the ratio $m:n$, then

$$P=\left(\frac{m{x}_2-n{x}_1}{m-n},\frac{m{y}_2-n{y}_1}{m-n},\frac{m{z}_2-n{z}_1}{m-n}\right),$$ provided $m\ne n$.

• Collinear points: Points lying on the same straight line. In space, three points are collinear if their position coordinates satisfy the condition of lying on one common line.

Worked Methods

1. Locating a point using coordinates

• Write the point as $P(x,y,z)$.
• Interpret $x$, $y$, and $z$ as signed coordinates measured relative to the $yz$-, $zx$-, and $xy$-planes.
• Determine the octant from the signs of $x$, $y$, and $z$.
• If one coordinate is zero, identify the corresponding coordinate plane:

- $z=0$: $xy$-plane; - $x=0$: $yz$-plane; - $y=0$: $zx$-plane.

• If two coordinates are zero, identify the relevant coordinate axis.

2. Finding distances from coordinate planes

For $P(x,y,z)$:

• Take the absolute value of the $x$-coordinate to find the distance from the $yz$-plane:

$$|x|.$$

• Take the absolute value of the $y$-coordinate to find the distance from the $zx$-plane:

$$|y|.$$

• Take the absolute value of the $z$-coordinate to find the distance from the $xy$-plane:

$$|z|.$$

3. Finding the distance between two points

Given $A(x_1,y_1,z_1)$ and $B(x_2,y_2,z_2)$:

• Find the differences between corresponding coordinates:

$$x_2-x_1,y_2-y_1,z_2-z_1.$$

• Square each difference.
• Add the three squared differences.
• Take the positive square root:

$$AB=\sqrt{(x_2-x_1{)}^2+(y_2-y_1{)}^2+(z_2-z_1{)}^2}.$$

4. Finding the midpoint

Given $A(x_1,y_1,z_1)$ and $B(x_2,y_2,z_2)$:

• Add the two $x$-coordinates and divide by $2$.
• Add the two $y$-coordinates and divide by $2$.
• Add the two $z$-coordinates and divide by $2$.
• Combine the results:

$$\text{Midpoint}=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2},\frac{z_1+z_2}{2}\right).$$

5. Applying the internal section formula

Suppose $P$ divides $A(x_1,y_1,z_1)$ and $B(x_2,y_2,z_2)$ internally in the ratio $m:n$.

• Confirm that the division is internal, so $P$ lies between $A$ and $B$.
• Use the denominator $m+n$.
• Apply the weighted-average formula separately to each coordinate:

$$x_P=\frac{m{x}_2+n{x}_1}{m+n},$$ $$y_P=\frac{m{y}_2+n{y}_1}{m+n},$$ $$z_P=\frac{m{z}_2+n{z}_1}{m+n}.$$

• Combine the three values:

$$P=\left(\frac{m{x}_2+n{x}_1}{m+n},\frac{m{y}_2+n{y}_1}{m+n},\frac{m{z}_2+n{z}_1}{m+n}\right).$$

6. Applying the external section formula

Suppose $P$ divides $A(x_1,y_1,z_1)$ and $B(x_2,y_2,z_2)$ externally in the ratio $m:n$.

• Confirm that $P$ lies outside the line segment.
• Check the required condition $m\ne n$.
• Use the denominator $m-n$.
• Apply the formula component-wise:

$$x_P=\frac{m{x}_2-n{x}_1}{m-n},$$ $$y_P=\frac{m{y}_2-n{y}_1}{m-n},$$ $$z_P=\frac{m{z}_2-n{z}_1}{m-n}.$$

• Combine the three values:

$$P=\left(\frac{m{x}_2-n{x}_1}{m-n},\frac{m{y}_2-n{y}_1}{m-n},\frac{m{z}_2-n{z}_1}{m-n}\right).$$

7. Checking collinearity

• Identify the three points in space.
• Test whether their position coordinates satisfy the condition of lying on one common straight line.
• Conclude that the points are collinear only if they lie on the same line.

Where It Goes Wrong

• Confusing the coordinate planes: the $xy$-plane requires $z=0$, the $yz$-plane requires $x=0$, and the $zx$-plane requires $y=0$.
• Omitting absolute values when calculating distances from coordinate planes; these distances are $|x|$, $|y|$, and $|z|$, not necessarily the signed coordinates.
• Applying a two-dimensional formula incompletely: distance, midpoint, and section formulas must be applied to all three coordinate components.
• Reversing the endpoint coordinates or the weights in the section formula; the order of $A$ and $B$ must be used consistently.
• Using the internal formula for external division, or vice versa; internal division uses $m+n$, whereas external division uses $m-n$.
• Forgetting the condition $m\ne n$ for external division.

What Gets Asked

• Represent a point in three-dimensional space using $P(x,y,z)$.
• Identify the coordinate plane or axis containing a point.
• Determine the octant from the signs of $x$, $y$, and $z$.
• Find the distances of a point from the $yz$-, $zx$-, and $xy$-planes.
• Calculate the distance between two points in space.
• Find the midpoint of a line segment joining two points.
• Apply the internal section formula for a given ratio $m:n$.
• Apply the external section formula, including the condition $m\ne n$.
• Determine whether three points are collinear.