LENGKAP MUDAH DIPAHAMI! Dimensi tiga KUBUS jarak titik ke garis

Introduction

This tutorial focuses on understanding the concept of distance from a point to a line in three-dimensional space, specifically within the context of a cube. This knowledge is essential for high school students (kelas 12) studying mathematics, particularly in accordance with the 2013 curriculum. By mastering these concepts, students can confidently tackle problems related to three-dimensional geometry.

Step 1: Understanding the Cube Structure

To effectively analyze distances in three-dimensional space, begin by familiarizing yourself with the structure of a cube.

• A cube has 8 vertices (corners), 12 edges, and 6 faces.
• Each edge of the cube is equal in length and meets at right angles.
• Visualize the cube in a coordinate system, with corners positioned at:
  • (0, 0, 0)
  • (0, 0, a)
  • (0, a, 0)
  • (0, a, a)
  • (a, 0, 0)
  • (a, 0, a)
  • (a, a, 0)
  • (a, a, a)

Here, 'a' represents the length of the cube’s edge.

Step 2: Identifying Points and Lines

Next, establish the points and lines relevant to your calculations.

• Identify the point from which you want to measure the distance. For example, let’s say we have point P located at (x₁, y₁, z₁).
• Determine the line you are measuring to, which can be represented by two points, A and B. For example, A (x₂, y₂, z₂) and B (x₃, y₃, z₃).

Step 3: Calculating the Vector

Transform the points into vectors to facilitate distance calculations.

• Create vectors from points:
  • Vector AP = (x₁ - x₂, y₁ - y₂, z₁ - z₂)
  • Vector AB = (x₃ - x₂, y₃ - y₂, z₃ - z₂)

Step 4: Finding the Cross Product

The cross product of vectors provides a way to find the perpendicular distance.

• Calculate the cross product ( AP \times AB ):
  • If AP = (a₁, b₁, c₁) and AB = (a₂, b₂, c₂), then:

  AP × AB = (b₁c₂ - c₁b₂, c₁a₂ - a₁c₂, a₁b₂ - b₁a₂)
  

Step 5: Calculating the Magnitude

To find the distance from point P to the line AB, first compute the magnitude of the cross product.

• The magnitude is given by:

|AP × AB| = √((b₁c₂ - c₁b₂)² + (c₁a₂ - a₁c₂)² + (a₁b₂ - b₁a₂)²)

Step 6: Distance Formula

Finally, apply the distance formula between point and line.

• The distance ( d ) is calculated as:

d = |AP × AB| / |AB|

• Where |AB| is the magnitude of vector AB:

|AB| = √((x₃ - x₂)² + (y₃ - y₂)² + (z₃ - z₂)²)

Conclusion

In this tutorial, you learned how to calculate the distance from a point to a line in three-dimensional space, using a cube as a reference. By understanding the structure of the cube, identifying relevant points and lines, and applying vector calculations, you can solve complex geometry problems effectively. Practice these steps with different points and lines to solidify your understanding, and consider exploring additional resources or problems related to three-dimensional geometry.