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Introduction

In this tutorial, we will explore the concept of distance from a point to a line in three-dimensional space. This is a fundamental topic in geometry, particularly useful in mathematics and engineering. We will calculate the distances from specific points to designated lines using clear steps that you can follow along with.

Step 1: Understanding the Concept of Distance from a Point to a Line

• The distance from a point to a line is the shortest distance between the point and any point on the line.
• In three-dimensional space, this can be visualized as a perpendicular line drawn from the point to the line.

Practical Tip

To visualize this, you can sketch the point and the line on graph paper, ensuring you draw the shortest distance accurately.

Step 2: Calculate Distance from Point B to Line AC

1. Identify the coordinates of point B and the endpoints of line AC.
   • Example coordinates:
     • Point B: (x1, y1, z1)
     • Line A: (xA1, yA1, zA1), Line C: (xA2, yA2, zA2)
2. Use the distance formula for a point to a line in 3D: [ D = \frac{|(P - A) \cdot (B - A)|}{|B - A|} ] where:
   • (P) is the point (B),
   • (A) and (B) are points on the line.

Common Pitfall

Ensure that you correctly apply the coordinates in the formula, as errors can lead to incorrect distance calculations.

Step 3: Calculate Distance from Point C to Line BG

1. Identify the coordinates for point C and the endpoints of line BG.
   • Example coordinates:
     • Point C: (x2, y2, z2)
     • Line B: (xB1, yB1, zB1), Line G: (xB2, yB2, zB2)
2. Apply the same distance formula as in Step 2.

Step 4: Calculate Distance from Point A to Lines HF and CH

1. For Line HF:
   • Identify coordinates for point A and endpoints of line HF.
   • Example coordinates:
     • Point A: (x3, y3, z3)
     • Line H: (xH1, yH1, zH1), Line F: (xH2, yH2, zH2)
2. For Line CH:
   • Identify coordinates for line CH and point A.
   • Apply the distance formula for both lines.

Practical Application

This method can be used in various fields, including physics for trajectory calculations and computer graphics for rendering distances in 3D environments.

Conclusion

In this tutorial, we have learned how to calculate the distance from points to lines in three-dimensional space. By following the steps outlined for each point and line combination, you can accurately determine these distances. For further practice, consider applying these techniques to different points and lines or explore related geometry topics.