Problem 46 - Thomas Calculus

Question

In Exercises \(41-46,\) find the distance between points \(P_{1}\) and \(P_{2}\) $$P_{1}(5,3,-2), \quad P_{2}(0,0,0)$$

Step-by-Step Solution

Answer

The distance between points is \( \sqrt{38} \).

Step 1: Understand the Formula

To find the distance between two points in a 3-dimensional space, we use the distance formula: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \] Where \((x_1, y_1, z_1)\) and \((x_2, y_2, z_2)\) are the coordinates of the points \(P_1\) and \(P_2\) respectively.

Step 2: Identify the Coordinates

The coordinates for point \(P_1\) are \((5, 3, -2)\), and for point \(P_2\), they are \((0, 0, 0)\). These values will be substituted into the distance formula.

Step 3: Substitute the Coordinates into the Formula

Substitute \(x_1 = 5\), \(y_1 = 3\), \(z_1 = -2\), \(x_2 = 0\), \(y_2 = 0\), and \(z_2 = 0\) into the distance formula:\[ d = \sqrt{(0 - 5)^2 + (0 - 3)^2 + (0 + 2)^2} \]

Step 4: Calculate Each Squared Difference

Calculate the squared differences: \((0 - 5)^2 = 25\)\((0 - 3)^2 = 9\)\((0 + 2)^2 = 4\)

Step 5: Add Squared Differences

Add the squared differences: \[ 25 + 9 + 4 = 38 \]

Step 6: Compute the Square Root

Finally, compute the square root of the sum to find the distance:\[ d = \sqrt{38} \]

Key Concepts

3D Geometry

Three-dimensional (3D) geometry is all about understanding spaces and forms that have depth, height, and width. In 3D, objects are not flat like they are in 2D; they have volume. This can be visualized in the world around us as boxes, balls, and pyramids. In mathematics, understanding 3D geometry involves being able to plot and manipulate points in space using a coordinate system which determines their exact position.
In this exploration, we're interested in how distances can be calculated between points spread out in the three-dimensional space. This helps in understanding spatial relations and is applicable in real-life scenarios such as navigating with maps and designing objects using CAD software.

Coordinate System

A coordinate system is a way to uniquely identify each location in a space with numbers. In 3D geometry, it’s a set of three numbers, each corresponding to the x, y, and z axes respectively.

• The x-axis is typically considered the horizontal axis.
• The y-axis represents the vertical axis in 2D, but in 3D, it’s often still seen as vertical in a spatial sense.
• The z-axis introduces depth into the picture, letting us specify how far into or out of a plane an object is.

Coordinates like (5, 3, -2) tell us exactly where a point is located in this 3D space. Using the coordinate system, we can link points and calculate distances by analyzing differences in their coordinate values.
This system underpins much of our spatial understanding and is fundamental in mathematics, physics, engineering, and various other fields wherever 3D positioning is essential.

Calculating Distance

To calculate the distance between two points in 3D space, we use a special formula — the 3D distance formula. This is an extension of the Pythagorean theorem:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \]Here’s how we use it:
First, find the difference between the respective coordinates of each point.

• Subtract the x-coordinates: \(x_2 - x_1\).
• Subtract the y-coordinates: \(y_2 - y_1\).
• Subtract the z-coordinates: \(z_2 - z_1\).

Square these results to eliminate negative values and then add them together. Finally, take the square root to return to the original scale of distance; this gives you the straight-line distance between the points.
Mastering this calculation not only aids in solving math problems but is also incredibly beneficial in fields like architecture, physics, and computer graphics, where spatial relationships are crucial.