Problem 31 - Advanced Engineering Mathematics

Question

Solve for the unknown. $$ P_{1}(x, 2,3), P_{2}(2,1,1) ; d\left(P_{1}, P_{2}\right)=\sqrt{21} $$

Step-by-Step Solution

Answer

The possible values for \( x \) are -2 and 6.

Step 1: Understanding the Problem

We are given two points in a 3-dimensional space, \( P_1(x, 2, 3) \) and \( P_2(2, 1, 1) \). The problem states that the distance between these two points is \( \sqrt{21} \). Our task is to find the value of \( x \) that makes this statement true.

Step 2: Distance Formula in 3D

The distance \( d \) between any two points \( (x_1, y_1, z_1) \) and \( (x_2, y_2, z_2) \) in three-dimensional space is given by the formula: \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \). We will apply this formula to the given points \( P_1(x, 2, 3) \) and \( P_2(2, 1, 1) \).

Step 3: Substitute the Points into the Formula

Substitute the coordinates of the points into the distance formula:\[ d = \sqrt{(2 - x)^2 + (1 - 2)^2 + (1 - 3)^2} \]Simplify the expressions inside the square root:\[ d = \sqrt{(2 - x)^2 + (-1)^2 + (-2)^2} \]Which further simplifies to:\[ d = \sqrt{(2 - x)^2 + 1 + 4} \]\[ d = \sqrt{(2 - x)^2 + 5} \]

Step 4: Set the Equation with Given Distance

According to the problem, the distance \( d = \sqrt{21} \). Therefore, we set up the equation:\[ \sqrt{(2 - x)^2 + 5} = \sqrt{21} \]To solve for \( x \), we will first square both sides to eliminate the square root.

Step 5: Solve for x

Square both sides of the equation:\[ (2 - x)^2 + 5 = 21 \]Subtract 5 from both sides:\[ (2 - x)^2 = 16 \]Take the square root of both sides:\[ 2 - x = \pm 4 \]This gives us two possible equations:1. \( 2 - x = 4 \)2. \( 2 - x = -4 \)

Step 6: Find Possible Values of x

Solve each equation for \( x \):1. For \( 2 - x = 4 \): - Subtract 2 from both sides: \( -x = 2 \). - Multiply by -1: \( x = -2 \).2. For \( 2 - x = -4 \): - Subtract 2 from both sides: \( -x = -6 \). - Multiply by -1: \( x = 6 \).Thus, the two possible values for \( x \) are \( x = -2 \) and \( x = 6 \).

Key Concepts

Solve for Unknown

To solve for the unknown value in our problem, we leverage the 3D distance formula to identify the variable in question. In this instance, we're working with two points, \(P_1(x, 2, 3)\) and \(P_2(2, 1, 1)\). We know that the distance between these points is \(\sqrt{21}\).

The main goal here is to find the value of \(x\) that makes the statement about the distance true. To do this, we first write the distance formula based on the point coordinates, which yields the equation:

• \(\sqrt{(2-x)^2 + (1-2)^2 + (1-3)^2} = \sqrt{21}\)

By simplifying this expression and solving the resulting equation for \(x\), we determine that \(x\) must be either \(-2\) or \(6\).

Solving for the unknown like this helps us validate the distances between points and ensures accuracy in determining the dimensions of a space.

Distance Between Points

The distance between two points in a three-dimensional space is essential in fields like physics and computer graphics. Here, we use the 3D distance formula: \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}\). It calculates how far apart two points are by considering all three axes of space: the x, y, and z dimensions.

In our example, given points \(P_1(x, 2, 3)\) and \(P_2(2, 1, 1)\), the formula helps us include each coordinate's contribution to the overall distance:

• Subtraction of x-coordinates: \((2 - x)^2\),
• Subtraction of y-coordinates: \((1 - 2)^2\),
• Subtraction of z-coordinates: \((1 - 3)^2\).

This method allows us to find precise measurements, so we know how objects are positioned or how paths travel through space.

Distance calculation in 3D isn't just a mathematical curiosity; it's a practical tool that finds application in design, technology, and natural sciences.

Three-dimensional Space

Three-dimensional space is our everyday world, where objects have width, height, and depth. When dealing with such spaces mathematically, it's vital to understand how to pinpoint locations with coordinates in three axes: x, y, and z.

In any given task, like finding the distance between two points, we use a system of coordinates to specify exact positions. In our problem, for instance, coordinates \(P_1(x, 2, 3)\) and \(P_2(2, 1, 1)\) help pinpoint the location of those points within this space. These coordinates explain not just where the points are, but also how far apart they are.

The way we navigate these spaces mathematically often involves a formulaic approach. You'll encounter the 3D distance formula frequently, as it links directly to how things move and interact in real world environments.

Understanding three-dimensional space through coordinate systems is foundational, not only in academia but also in everyday tasks like mapping and even virtual reality environments.