6. The 3-dimensional Co-ordinate System

We can expand our 2-dimensional (x-y) coordinate system into a 3-dimensional coordinate system, using x-, y-, and z-axes.

3D- axes

The x-y plane is horizontal in our diagram above and shaded green. It can also be described using the equation z = 0, since all points on that plane will have 0 for their z-value.

The x-z plane is vertical and shaded pink above. This plane can also be described using the equation y = 0.

The y-z plane is also vertical and shaded blue. The y-z plane can also be described using the equation x = 0.

We normally use the 'right-hand orientation' for the 3 axes, with the the positive x-axis pointing in the direction of the first finger of our right hand, the positive y-axis pointing in the direction of our second finger and the positive z-axis pointing up in the direction of our thumb.

right hand rule

Example - Points in 3-D Space

In 3-dimensional space, the point (2, 3, 5) is graphed as follows:

(2,3,5)

To reach the point (2, 3, 5), we move 2 units along the x-axis, then 3 units in the y-direction, and then up 5 units in the z-direction.

Distance in 3-dimensional Space

To find the distance from one point to another in 3-dimensional space, we just extend Pythagoras' Theorem.

Distance from the Origin

The general point P (a, b, c) is shown on the 3D graph below. The point N is directly below P on the x-y plane.

distance

The distance from (0, 0, 0) to the point P (a, b, c) is given by:

distance OP = √ (a² + b² + c²)

Why?

The point N (a, b, 0) is shown on the graph. From Pythagoras' Theorem,

distance ON = √ (a² + b²)

and squaring both sides gives:

(ON)² = a² + b²

Distance NP is simply c (this is the distance up the z-axis for the point P).

Applying Pythagoras' Theorem for the triangle ONP, we have:

distance OP

= √ ( (ON)² + c²)

= √(a² + b² + c²)

Example - Distance from the Origin to a Point

Find the distance from the origin O to the point B (2, 3, 5). This is the example from above.

Answer

Distance Between 2 Points in 3 Dimensions

If we have point A (x₁, y₁, z₁) and another point B (x₂, y₂, z₂) then the distance AB between them is given by the formula:

distance AB = √[(x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²]

This is just an extension of the distance formula (from the origin to a point) that we met above.

Example

Find the distance between the points P (2, 3, 5) and Q (4, -2, 3).

Answer

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