Problem 44 - Thomas Calculus

Question

In Exercises \(41-46,\) find the distance between points \(P_{1}\) and \(P_{2}\) $$P_{1}(3,4,5), \quad P_{2}(2,3,4)$$

Step-by-Step Solution

Answer

The distance is \( \sqrt{3} \).

Step 1: Understand the Problem

We need to find the distance between two points in a 3-dimensional space, given their coordinates. These points are \( P_1(3, 4, 5) \) and \( P_2(2, 3, 4) \).

Step 2: Recall the Distance Formula

In 3D space, the distance \( d \) between two points \( (x_1, y_1, z_1) \) and \( (x_2, y_2, z_2) \) is given by the formula:\[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}\]

Step 3: Substitute the Coordinates

Substitute the coordinates of \( P_1 \) and \( P_2 \) into the distance formula: \[d = \sqrt{(2 - 3)^2 + (3 - 4)^2 + (4 - 5)^2}\]

Step 4: Calculate the Differences

Calculate the differences of each coordinate: - For \( x \): \( 2 - 3 = -1 \) - For \( y \): \( 3 - 4 = -1 \) - For \( z \): \( 4 - 5 = -1 \).

Step 5: Square the Differences

Square each of the differences to eliminate negative signs:- \((-1)^2 = 1\)- \((-1)^2 = 1\)- \((-1)^2 = 1\).

Step 6: Sum the Squares

Add the squared differences together: \[1 + 1 + 1 = 3\]

Step 7: Find the Square Root

Take the square root of the sum from Step 6 to find the distance:\[d = \sqrt{3}\]

Step 8: Conclusion

The distance between points \( P_1(3, 4, 5) \) and \( P_2(2, 3, 4) \) is \( \sqrt{3} \).

Key Concepts

3D Geometry

Three-dimensional geometry is the branch of mathematics that deals with objects in 3-dimensional space. Unlike two-dimensional geometry, which we commonly encounter with shapes on a flat surface, 3D geometry involves objects that have depth in addition to height and width.
Imagine a room where you can move not just along the floor but also up and down. This extra dimension adds a level of complexity and fascination. For students, understanding 3D geometry introduces a new way to visualize space and how objects relate to each other in that space.
Some important characteristics of 3D space include:

• Three coordinate axes: x, y, and z
• Geometric objects like points, lines, and planes
• Physical properties such as volume and surface area

In our exercise, 3D geometry provides the framework to calculate the distance between two points, making it fundamental to problems involving multiple dimensions.

Coordinate System

The coordinate system in 3D geometry allows us to locate any point in space using a set of three numbers. Each number corresponds to a point's position along one of the three-dimensional axes (x, y, and z). This system is fundamental to graphing and analyzing geometric problems in three dimensions.
In a typical 3D coordinate system:

• The x-axis runs left to right.
• The y-axis runs front to back.
• The z-axis runs up and down.

By using these three coordinates, you can precisely describe the location of any point in space. For example, the points from our exercise,
- For \( P_1(3, 4, 5) \), 3 is the x-coordinate, 4 is the y-coordinate, and 5 is the z-coordinate.
- For \( P_2(2, 3, 4) \), 2, 3, and 4 are the x, y, and z coordinates respectively.
This method straightforwardly extends our understanding from 2D to 3D, and is essential for distance calculations and other geometric assessments in three-dimensional space.

Distance Calculation

Finding the distance between two points in 3D space might first seem challenging, but it builds on concepts you might already know from 2D geometry. A well-known formula comes to our aid, enabling this calculation seamlessly by extending the Pythagorean Theorem into three dimensions.

To calculate the distance between two points, say \(P_1(x_1, y_1, z_1)\) and \(P_2(x_2, y_2, z_2)\), you can apply the 3D distance formula:\[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}\]

Understanding the Calculation Steps:

1. **Identify the coordinate differences:** Begin by determining how much each coordinate varies between the two points.2. **Square each difference:** Squaring each difference ensures all values are positive, reflecting true separation in space.3. **Sum the squares:** Combine these squared values, a key step that echoes combining lengths of sides within a triangle as per the Pythagorean theorem.4. **Take the square root:** This step transfers the summed value into the final distance measurement.For our specific example, once all calculations are complete, the distance between \(P_1(3, 4, 5)\) and \(P_2(2, 3, 4)\) emerges as \(\sqrt{3}\), a concise result of combining these straightforward mathematical operations.