Chapter Contents

• Introduction to Vectors
• 1. Vector Concepts and Notation
• 2. Vector Addition in 1-D
• 3. Vectors in 2 Dimensions
• 4. Adding Vectors in 2-D
• Adding Vectors using SVG graphs
• 5. Dot Product of 2-D Vectors
• 6. Three-dimensional Space
• 3-D and Contour Grapher
• 7. Vectors in 3-dimensional Space
• 8. Cross Product of 2 Vectors
• 9. Variable Vectors
• 10. Vector Calculus
• 3-D Earth Geometry
• Vector Art
• Introduction to Vectors Problem Solver

• Comments, Questions?

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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.

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 be described using the equation y = 0.

The y-z plane is also vertical and shaded blue. The y-z plane can 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.

Example - Points in 3-D Space

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

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.

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 1 - 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 2 - Distance between 2 points

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

Answer