Three-dimensional space

Coordinates in three dimensions, right-handed axes, the coordinate planes, distances and midpoints in space, and the equation of a sphere.

Coordinates in space

Three mutually perpendicular axes meet at the origin. They are drawn right-handed: curl the fingers of your right hand from the positive $x$-axis to the positive $y$-axis, and your thumb points along the positive $z$-axis. Each point has coordinates $(x, y, z)$.

The three coordinate planes are the $xy$-plane ($z = 0$), the $yz$-plane ($x = 0$) and the $xz$-plane ($y = 0$). More generally, $z = 3$ is the horizontal plane $3$ units above the $xy$-plane.

Distance and midpoint

In three dimensions

$$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \qquad M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \frac{z_1 + z_2}{2}\right)$$

The distance formula comes from Pythagoras' theorem used twice, as for the diagonal of a rectangular box.

Example 1●

Find the distance from $A(1, -2, 3)$ to $B(3, 2, -1)$, and the midpoint of $AB$.

$d = \sqrt{2^2 + 4^2 + (-4)^2} = \sqrt{36} = 6$.

$M = (2, 0, 1)$.

Example 2: a box●●

Find the length of the longest diagonal of a $2 \times 3 \times 6$ box.

$\sqrt{2^2 + 3^2 + 6^2} = \sqrt{49} = 7$.

Spheres

Sphere with centre (a, b, c) and radius r

$$(x - a)^2 + (y - b)^2 + (z - c)^2 = r^2$$

Example 3●●

Find the centre and radius of $x^2 + y^2 + z^2 - 4x + 6z = 3$.

Complete the squares: $(x - 2)^2 + y^2 + (z + 3)^2 = 3 + 4 + 9 = 16$.

Centre $(2, 0, -3)$, radius $4$.

Flashcards

Distance in 3D

$\sqrt{\Delta x^2 + \Delta y^2 + \Delta z^2}$

The $xy$-plane

$z = 0$

Sphere

$(x - a)^2 + (y - b)^2 + (z - c)^2 = r^2$

Practice

Find the distance from $A(1, -2, 3)$ to $B(3, 2, -1)$.
Find the midpoint of the same points. Enter it as $x, y, z$.
Find the length of the longest diagonal of a $2 \times 3 \times 6$ box.
Find the distance from the origin to $(1, 2, 2)$.
Find the radius of $x^2 + y^2 + z^2 - 4x + 6z = 3$.
Find the distance from $(3, 4, 7)$ to the $xy$-plane.
The $yz$-plane has equation:
The sphere $(x - 1)^2 + (y + 2)^2 + z^2 = 9$ has centre:
In right-handed axes, curling the fingers from the positive $x$-axis to the positive $y$-axis makes the thumb point along: