To find the distance between points (x_1, y_1) and (x_2, y_2) in two dimensions we use the formula:

d

=

\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

In three dimensions, we need to add a third variable, z. This adds a third dimension to the Cartesian plane as shown:

3 dimensional Cartesian plane

A point on the 3D plane has coordinates: (x, y, z).

Remember!

So to find the distance between (x_1, y_1, z_1) and (x_2, y_2, z_2) on the 3D plane we would use the following formula:

d

=

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

Worked examples

Example 1

Find the distance between the points (3, 0, -2) and (4, 5, -6).

Think: We need to substitute these coordinates into the above formula. You can choose which point you let be (x_1, y_1, z_1) and (x_2, y_2, z_2). In the working, we have let (x_1, y_1, z_1) = (3, 0, -2), and (x_2, y_2, z_2) = (4, 5, -6).

Do:

d	=	\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}
	=	\sqrt{(4 - 3)^2 + (5 - 0)^2 + (-6 - (-2))^2}
	=	\sqrt{1^2 + 5^2 + (-4)^2}
	=	\sqrt{1 + 25 + 16}
	=	\sqrt{42}
	\approx	6.48

So the distance between the two points is approximately 6.48 units.

Example 2

Find the exact distance between the two points shown in the 3D Cartesian plane:

Think: First we should let (x_1, y_1, z_1) = (2, -5, 3), and (x_2, y_2, z_2) = (3, 3, -2).

Do:

d	=	\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}
	=	\sqrt{(3 - 2)^2 + (3 - (-5))^2 + (-2 - 3)^2}
	=	\sqrt{1^2 + 8^2 + (-5)^2}
	=	\sqrt{1 + 25 + 16}
	=	\sqrt{90}
	=	3\sqrt{10}

So the exact distance between the two points is 3\sqrt{10} units.

Midpoint of two points in 3D

We can also find the midpoint of two points in the 3D plane by using the z-coordinate.

Remember!

So to find the midpoint between (x_1, y_1, z_1) and (x_2, y_2, z_2) on the 3D plane we would use the following formula:

Midpoint

=

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

Example 3

Find the coordinates of the midpoint between the points shown in the diagram:

Think: First we should let (x_1, y_1, z_1) = (1, 6, 0), and (x_2, y_2, z_2) = (-2, 5, 4).

Do:

Midpoint	=	\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \frac{z_1 + z_2}{2} \right)
	=	\left( \frac{1 + (-2)}{2}, \frac{6 + 5}{2}, \frac{0 + 4}{2} \right)
	=	\left( \frac{-1}{2}, \frac{11}{2}, 2 \right)

3.04 Two points in 3D

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