Problem 26 - University Calculus: Early Transcendentals

Question

Find the distance between points \(P_{1}\) and \(P_{2}\) $$P_{1}(-1,1,5), \quad P_{2}(2,5,0)$$

Step-by-Step Solution

Answer

The distance is \(5\sqrt{2}\).

Step 1: Identify Coordinates

First, we identify the coordinates of the given points. Point \(P_1\) has coordinates \((-1, 1, 5)\) and point \(P_2\) has coordinates \((2, 5, 0)\).

Step 2: Apply Distance Formula

We use the distance formula for three-dimensional space between two points \((x_1, y_1, z_1)\) and \((x_2, y_2, z_2)\) defined as: \ \[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}\] \

Step 3: Plug in the Coordinates

Substitute the coordinates of \(P_1\) and \(P_2\) into the formula: \ \[d = \sqrt{(2 - (-1))^2 + (5 - 1)^2 + (0 - 5)^2}\]

Step 4: Simplify Inside the Square Root

Calculate the differences and their squares: \ \[d = \sqrt{(3)^2 + (4)^2 + (-5)^2}\] \ which simplifies to \ \[d = \sqrt{9 + 16 + 25}\]

Step 5: Final Calculation

Sum the values inside the square root and compute the final value: \ \[d = \sqrt{50}\] \ This can be further simplified to \ \[d = 5\sqrt{2}\].

Key Concepts

three-dimensional space

Three-dimensional space is a concept that extends the idea of the two-dimensional plane into a higher dimension. In three-dimensional space, any point can be identified using three coordinates, each representing a measurement along one of the space's three axes: x, y, and z. This space is often visualized with the Cartesian coordinate system, which is a mathematical representation that helps in locating points within this space.
Visualize this like a room where:

• The x-axis measures horizontal distance.
• The y-axis measures vertical distance.
• The z-axis measures depth, adding another layer compared to the flat surface of the two-dimensional plane.

Understanding three-dimensional space is crucial for tasks involving width, height, and depth, and it forms the foundation for fields like physics, engineering, and computer graphics.

coordinate geometry

Coordinate geometry, also known as analytic geometry, focuses on using algebraic symbols and equations to describe geometric figures on a coordinate plane. This branch of mathematics allows us to explore the relationships and properties of geometric shapes using algebraic formulas.
In three-dimensional space, coordinate geometry involves using the coordinates (x, y, z) to identify and study points, lines, and surfaces. Here are a few key points about coordinate geometry:

• It provides a link between algebra and geometry, making it easier to calculate distances, angles, and areas.
• The system coordinates are used to perform operations to solve problems and prove theorems.
• The distance formula, slope formula, and equations of planes and lines are common examples of tools used in coordinate geometry.

This approach greatly simplifies complex spatial problems by converting them into solvable algebraic equations.

distance between points

The distance between points in three-dimensional space can be calculated using the distance formula, which is an extension of the Pythagorean theorem. This formula allows us to find the length of the straight line connecting two points with known coordinates in space.
The general distance formula in three dimensions between two points \(x_1, y_1, z_1\) and \(x_2, y_2, z_2\) is:
\[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}\]
This helps to break down the calculation into simpler parts involving subtraction, squaring, and taking the square root, covering:

• The difference along the x-axis, \(x_2 - x_1\).
• The difference along the y-axis, \(y_2 - y_1\).
• The difference along the z-axis, \(z_2 - z_1\).

Each squared difference represents a piece of the total length, combined to form the squared sides of a right triangle. The resulting square root gives the direct linear distance between the points, offering a precise measure regardless of their positions in three-dimensional space.