Problem 39 - Thomas Calculus

Question

In Exercises \(35-40,\) find the distance between points \(P_{1}\) and \(P_{2}\) $$ P_{1}(0,0,0), \quad P_{2}(2,-2,-2) $$

Step-by-Step Solution

Answer

The distance between the points is \(2\sqrt{3}\).

Step 1: Identify the Coordinates

The coordinates of point \( P_1 \) are \((0, 0, 0)\) and the coordinates of point \( P_2 \) are \((2, -2, -2)\).

Step 2: Use the Distance Formula

The distance \( d \) between two points in 3-dimensional space, \((x_1, y_1, z_1)\) 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

Plug the coordinates of \( P_1 \) and \( P_2 \) into the distance formula: \( d = \sqrt{(2-0)^2 + (-2-0)^2 + (-2-0)^2} \).

Step 4: Simplify the Expression

Calculate the difference for each pair of coordinates and simplify: \( d = \sqrt{2^2 + (-2)^2 + (-2)^2} = \sqrt{4 + 4 + 4} \).

Step 5: Calculate the Distance

Add the values under the square root and then compute the square root: \( d = \sqrt{12} = 2\sqrt{3} \).

Key Concepts

3-dimensional space

In mathematics, three-dimensional space refers to a geometric setting in which three values are required to determine the position of an element (i.e., a point). Just like how two-dimensional space uses an x-axis and y-axis, three-dimensional space adds a z-axis. This extra dimension allows us to represent any location within our world, which we often think of as having length, width, and height. In

• Physics, we use it to model the physical universe.
• Engineering, for creating 3D models.
• Computer graphics, in the design of realistic video game environments.

This space enables us to perform complex calculations, such as finding the distance between two points at different elevations, which is essential in fields like architecture and astronomy. The coordinates in a 3D space are represented as extit{x, y, z}, which indicates their position along each axis.
Understanding this concept helps us apply geometric principles beyond just flat shapes to more practical, real-world volumes and forms.

coordinate geometry

Coordinate geometry, also known as analytic geometry, merges algebra and geometry by using a coordinate system to define geometric shapes. This branch of geometry allows us to perform calculations on geometric objects by representing them with numerical coordinates. For example, in 2D, a point is defined by two coordinates, extit{x} for horizontal position and extit{y} for vertical. In 3D, we include a third coordinate: extit{z} for depth.

• This approach simplifies the analysis of geometric shapes.
• Facilitates solving complex geometric problems using algebraic equations.
• Helps in visualizing geometrical objects and measuring distances easily.

By using algebra, you can find distances, angles, and even work with shapes like circles and lines in three-dimensional space. It becomes possible to accurately calculate properties of shapes, such as area, volume, and centroid, by making use of the relationship between geometry and algebra. This is fundamental when dealing with multi-dimensional problems in diverse disciplines such as engineering and computer science.

Euclidean distance

Euclidean distance is a concept from mathematics that measures the straight-line distance between two points in Euclidean space. It's the most basic form of distance and is widely used in various fields like computer science, physics, and machine learning. The formula for Euclidean distance in 2D space is \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\). In 3D space, we extend this to include the z-coordinate: \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}\).

• It's derived from the Pythagorean theorem.
• Offers a straightforward method to compare distances in various dimensions.
• Essential in algorithms that require similarity measurements.

This formula is pivotal for many practical applications, as it allows for the clear and precise computation of distance metrics in physical and digital spaces, helping ensure accuracy in processes like navigation and imaging technology.