Introduction to Three Dimensional Geometry

The One Thing

Three-dimensional geometry extends the Cartesian coordinate system by introducing three mutually perpendicular axes, so every point in space is represented by an ordered triple $(x,y,z)$. Spatial relationships such as distances, midpoints, section points, and positions relative to axes, planes, and octants can then be determined algebraically.

Definitions and Results

• Three-dimensional coordinate system: A system that locates points in space using three perpendicular coordinate axes: the $x$-axis, $y$-axis, and $z$-axis.
• Origin: The common point of the three coordinate axes, represented by $O(0,0,0)$.
• Coordinates of a point: A point is written as $P(x,y,z)$, where $x$, $y$, and $z$ are the directed distances of the point from the $yz$-plane, $zx$-plane, and $xy$-plane, respectively. Equivalently, the coordinates measure directed distances parallel to the $x$-, $y$-, and $z$-axes.
• Coordinate axes: The $x$-axis, $y$-axis, and $z$-axis are the three mutually perpendicular lines used to measure position in space.
• Coordinate planes:

- The $xy$-plane has equation $z=0$. - The $yz$-plane has equation $x=0$. - The $zx$-plane has equation $y=0$.

• Octants: The three coordinate planes divide space into eight regions called octants. The signs of $x$, $y$, and $z$ identify the octant. In standard ordering, the sign combinations are

$$(+,+,+),(-,+,+),(-,-,+),(+,-,+),(+,+,-),(-,+,-),(-,-,-),(+,-,-).$$

• Distance between two points: For $P(x_1,y_1,z_1)$ and $Q(x_2,y_2,z_2)$,

$$PQ=\sqrt{(x_2-x_1{)}^2+(y_2-y_1{)}^2+(z_2-z_1{)}^2}.$$ This distance is always non-negative.

• Distance from the origin: For $P(x,y,z)$,

$$OP=\sqrt{x^2+y^2+z^2}.$$

• Section formula: If $P$ divides the line segment joining two points in a specified ratio, its coordinates are obtained by weighted averages or differences of the endpoint coordinates.
• Midpoint: The point dividing a line segment into two equal parts. For $A(x_1,y_1,z_1)$ and $B(x_2,y_2,z_2)$, the midpoint is

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

• Distance from coordinate planes: For $P(x,y,z)$, the distances from the $yz$-plane, $zx$-plane, and $xy$-plane are respectively

$$|x|,|y|,|z|.$$

• Points on coordinate axes:

- $P$ lies on the $x$-axis if $y=0$ and $z=0$. - $P$ lies on the $y$-axis if $x=0$ and $z=0$. - $P$ lies on the $z$-axis if $x=0$ and $y=0$.

• Points on coordinate planes:

- $P$ lies on the $xy$-plane if $z=0$. - $P$ lies on the $yz$-plane if $x=0$. - $P$ lies on the $zx$-plane if $y=0$.

Worked Methods

1. Representing a point in three-dimensional space

• Write the point as an ordered triple:

$$P(x,y,z).$$

• Interpret $x$, $y$, and $z$ as directed distances parallel to the corresponding axes.
• Use the zero coordinates and signs to determine the point’s position:

- zero coordinates identify coordinate axes or planes; - signs identify the octant.

The coordinate system extends the two-dimensional Cartesian system by adding a third perpendicular axis.

2. Finding the distance between two points

For $P(x_1,y_1,z_1)$ and $Q(x_2,y_2,z_2)$:

• Find the coordinate differences:

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

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

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

The formula follows by applying the Pythagorean theorem successively in perpendicular directions.

3. Finding the distance from the origin

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

• Use the origin $O(0,0,0)$.
• Substitute the coordinates into the distance formula:

$$OP=\sqrt{(x-0{)}^2+(y-0{)}^2+(z-0{)}^2}.$$

• Simplify:

$$OP=\sqrt{x^2+y^2+z^2}.$$

4. Finding the midpoint

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

• Add the corresponding $x$-coordinates, $y$-coordinates, and $z$-coordinates.
• Divide each sum by $2$:

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

• The resulting point divides $AB$ into two equal parts.

5. Applying the internal section formula

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

• Multiply the coordinates of $B$ by $m$.
• Multiply the coordinates of $A$ by $n$.
• Add the corresponding weighted coordinates.
• Divide each result by $m+n$:

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

The formula gives a weighted average located between $A$ and $B$.

6. Applying the external section formula

If $P$ divides $AB$ externally in the ratio $m:n$:

• Multiply the coordinates of $B$ by $m$.
• Multiply the coordinates of $A$ by $n$.
• Subtract the corresponding weighted coordinates.
• Divide each result by $m-n$:

$$P=\left(\frac{m{x}_2-n{x}_1}{m-n},\frac{m{y}_2-n{y}_1}{m-n},\frac{m{z}_2-n{y}_1}{m-n}\right).$$ More precisely, the $z$-coordinate is $$\frac{m{z}_2-n{z}_1}{m-n},$$ so the complete formula is $$P=\left(\frac{m{x}_2-n{x}_1}{m-n},\frac{m{y}_2-n{y}_1}{m-n},\frac{m{z}_2-n{z}_1}{m-n}\right).$$

• The formula requires $m\ne n$, since division by $m-n$ must be defined.

7. Determining distances from coordinate planes

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

• To find the distance from the $yz$-plane, use $|x|$.
• To find the distance from the $zx$-plane, use $|y|$.
• To find the distance from the $xy$-plane, use $|z|$.

The absolute values are required because distances are non-negative, regardless of the signs of the coordinates.

8. Identifying axes, planes, and octants

• Examine which coordinates are zero:

- $y=z=0$: $x$-axis; - $x=z=0$: $y$-axis; - $x=y=0$: $z$-axis.

• Examine a single zero coordinate:

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

• If none of the coordinates is zero, examine the signs of $(x,y,z)$ to identify one of the eight octants.

Where It Goes Wrong

• Forgetting that a point must be written as an ordered triple $P(x,y,z)$, with the coordinate order corresponding to the $x$-, $y$-, and $z$-axes.
• Using the wrong coordinate-plane condition: the $xy$-plane requires $z=0$, the $yz$-plane requires $x=0$, and the $zx$-plane requires $y=0$.
• Omitting one of the three squared coordinate differences in the distance formula.
• Giving a negative distance by failing to use the positive square root or the absolute values $|x|$, $|y|$, and $|z|$ for distances from coordinate planes.
• Interchanging $m$ and $n$ in the section formulas; the internal formula uses weighted addition, whereas the external formula uses weighted subtraction.
• Applying the external section formula when $m=n$, even though it requires $m\ne n$.

What Gets Asked

This material supports questions that require students to:

• represent points in three-dimensional coordinates and interpret directed distances;
• identify the origin, coordinate axes, coordinate planes, and octants;
• determine whether a point lies on an axis or coordinate plane;
• identify an octant from the signs of $(x,y,z)$;
• calculate the distance between two points;
• calculate the distance of a point from the origin;
• find the midpoint of a line segment;
• find a point dividing a line segment internally or externally in a given ratio;
• calculate distances from the $yz$-plane, $zx$-plane, and $xy$-plane;
• explain how the three-dimensional distance formula follows from successive applications of the Pythagorean theorem.