Problem 27 - University Calculus: Early Transcendentals

Question

Find the distance between points \(P_{1}\) and \(P_{2}\) $$P_{1}(1,4,5), \quad P_{2}(4,-2,7)$$

Step-by-Step Solution

Answer

The distance between the points is 7 units.

Step 1: Understand the Problem

We need to find the distance between two points in three-dimensional space. The points are given as coordinates: \( P_{1}(1,4,5) \) and \( P_{2}(4,-2,7) \).

Step 2: Identify the Distance Formula

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

Step 3: Substitute the Coordinates

Identify the coordinates from the given data: \( x_1 = 1, y_1 = 4, z_1 = 5 \) and \( x_2 = 4, y_2 = -2, z_2 = 7 \). Substitute these into the distance formula: \[ d = \sqrt{(4 - 1)^2 + (-2 - 4)^2 + (7 - 5)^2} \]

Step 4: Simplify the Expression

Calculate each component inside the square root: \((4 - 1)^2 = 3^2 = 9\)\((-2 - 4)^2 = (-6)^2 = 36\)\((7 - 5)^2 = 2^2 = 4\). Add these results together: \( 9 + 36 + 4 = 49 \).

Step 5: Calculate the Square Root

Take the square root of the sum: \[ d = \sqrt{49} = 7 \].

Key Concepts

3D distance formula

In three-dimensional space, the 3D distance formula is a mathematical tool used to find the straight-line distance between two points. It's an extended version of the Pythagorean theorem that works for three dimensions. This formula is particularly useful in fields like geometry, physics, and computer graphics, where accurate spatial calculations are essential.

The distance formula for 3D space is expressed as:

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

This equation measures the straight path connecting two points: \( P_1(x_1, y_1, z_1) \) and \( P_2(x_2, y_2, z_2) \).
- The terms \( (x_2 - x_1), (y_2 - y_1), (z_2 - z_1) \) represent the differences in coordinates along each axis.
- By squaring these differences, we account for direction and distance separately.
- Finally, taking the square root gives the actual length from one point to another.

Coordinate geometry

Coordinate geometry, sometimes called analytic geometry, is a method of using algebra to solve geometric problems. By representing geometric shapes with equations and coordinates in a plane or space, complex calculations become manageable using algebra and simple arithmetic operations. This approach is highly valuable for solving problems that would be cumbersome or impossible to address through classical geometry alone.

In three-dimensional coordinate geometry:

• Each point is defined by three numbers, called coordinates: \( (x, y, z) \).
• The x, y, and z values describe the position along the respective axes in a Cartesian system.
• Using this system, we can systematically study forms and distances, such as the distance between two points.

Understanding the concept of coordinates lets us visualize spatial relationships, identify distances, and reference objects accurately in space, all crucial for further learning in physics, engineering, and technology.

Distance calculation steps

Calculating the distance between points in 3D space involves a structured sequence of steps, ensuring accuracy in the result. Here's a breakdown of the steps used in the process:

**Recognize the Points:** Identify the two points in 3D space for which you need to find the distance. In this case, they are \( P_1(1,4,5) \) and \( P_2(4,-2,7) \).

• This step ensures that you are clear about the input values.

**Apply the Distance Formula:** Use the formula \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \) to calculate the distance.

• Substitute values from your identified points into the formula.
• Compute each squared difference separately, as in \( (4 - 1)^2, (-2 - 4)^2, \text{ and } (7 - 5)^2 \).

**Sum the Squares and Compute the Root:** Add the calculated squares to get a single value to simplify under the square root. Lastly, take the square root:

• Adding values: \( 9 + 36 + 4 = 49 \).
• Final distance: \( \sqrt{49} = 7 \).
• The result is the length directly between the two points in the 3D space.

Following these steps methodically ensures you arrive at the correct distance, making it easier to manage and understand complex spatial relationships.