Points and Vectors in Three-Space

Leaving the flat page

A drone hovering over a field, an airplane on approach to Pearson airport and a satellite in orbit all need three numbers to say where they are: two for the position on a map and one for height. Forces on a building act in three dimensions too. In this lesson you extend everything you know about vectors to three-space. The big idea is simple: a point and a vector in three-space are both written as an ordered triple of numbers.

The three-dimensional coordinate system

Take three number lines, the x-, y- and z-axes, that meet at the origin O and are mutually perpendicular (each one is at 90° to the other two).

On paper we draw the x-axis coming toward you (slanting down-left), the y-axis to the right and the z-axis up. This is a right-handed system: point the fingers of your right hand along the positive x-axis and curl them toward the positive y-axis; your thumb points along the positive z-axis. Math, physics and computer graphics almost always use right-handed axes, and the Dot and Cross Products unit depends on it.

A point is written P(x, y, z). To plot P(2, 3, 4), start at O, move 2 units along the x-axis, then 3 units parallel to the y-axis, then 4 units up. Drawing the rectangular box with O and P as opposite corners makes the picture much easier to read.

Planes and octants

The axes, taken two at a time, determine three coordinate planes:

• the xy-plane, where z = 0 (the "floor"),
• the xz-plane, where y = 0,
• the yz-plane, where x = 0.

These three planes split space into eight octants, according to the signs of x, y and z. The octant where all three are positive is called the first octant. A point with exactly one zero coordinate lies in a coordinate plane, and a point with two zero coordinates lies on an axis. For example, (3, 0, −2) is in the xz-plane and (0, 0, 5) is on the z-axis.

Vectors in three-space

The position vector of P(a, b, c) is 𝐎𝐏 = [a, b, c]. Just as in two-space:

• the vector from A(x₁, y₁, z₁) to B(x₂, y₂, z₂) is 𝐀𝐁 = [x₂ − x₁, y₂ − y₁, z₂ − z₁] (head minus tail);
• two vectors are equal exactly when all three components are equal.

Notice that the same ordered triple can describe a point (a location) or a vector (a displacement). The curriculum asks you to recognize this: (2, 3, 4) is a place, while [2, 3, 4] is the arrow from O to that place, or any arrow equal to it.

Magnitude and distance

Use the Pythagorean theorem twice. Along the floor of the box, the diagonal d satisfies d² = a² + b². That diagonal is perpendicular to the vertical edge c, so

|[a, b, c]| = √(a² + b² + c²)

and the distance between A and B is |𝐀𝐁| = √((x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²).