TR-12: Distance Between Points in Space (Trigonometry series by Dennis F. Davis)

Introduction

In this tutorial, we will explore how to determine the distance between two points in three-dimensional space using an extension of the general distance formula. This concept is a crucial application of the Pythagorean Theorem and is essential for fields such as geometry, physics, and computer graphics.

Step 1: Understand the Distance Formula in 3D

To calculate the distance between two points in 3D space, we use the distance formula, which is an extension of the Pythagorean Theorem. The formula is given as:

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

Where:

• (d) is the distance between the two points.
• ((x_1, y_1, z_1)) are the coordinates of the first point.
• ((x_2, y_2, z_2)) are the coordinates of the second point.

Practical Tips

• Ensure that you correctly identify the coordinates of both points before substituting them into the formula.
• Remember that squaring a negative number will yield a positive result, which is crucial for this calculation.

Step 2: Identify the Coordinates

Select the two points you want to calculate the distance between. For example:

• Point A: ((x_1, y_1, z_1) = (2, 3, 5))
• Point B: ((x_2, y_2, z_2) = (6, 7, 9))

Common Pitfalls

• Double-check that you are inputting the correct coordinates into the formula. Errors in coordinates will lead to incorrect calculations.

Step 3: Substitute the Coordinates into the Formula

Now, substitute the coordinates of both points into the distance formula:

1. Calculate the differences:

   • (x_2 - x_1 = 6 - 2 = 4)
   • (y_2 - y_1 = 7 - 3 = 4)
   • (z_2 - z_1 = 9 - 5 = 4)

2. Substitute these values into the formula: [ d = \sqrt{(4)^2 + (4)^2 + (4)^2} ]

Step 4: Calculate the Distance

Now perform the calculations:

1. Square the differences:

   • ((4)^2 = 16)

2. Add the squared values: [ d = \sqrt{16 + 16 + 16} = \sqrt{48} ]

3. Simplify: [ d = 4\sqrt{3} \approx 6.93 ]

Real-World Applications

• This distance calculation is used in computer graphics to determine the distance between objects.
• In physics, it can be employed to calculate trajectories and distances in three-dimensional space.

Conclusion

In this tutorial, you learned how to apply the distance formula in three-dimensional space to find the distance between two points. Remember to follow each step methodically to avoid common mistakes, and practice with different sets of coordinates to reinforce your understanding. Next, consider exploring further applications of trigonometry in three dimensions or tackling the next video in the series for more advanced concepts.