Problem 76 - University Calculus: Early Transcendentals

Question

\(.\) Find all points that simultaneously lie 3 units from each of the points \((2,0,0),(0,2,0),\) and (0,0,2)

Step-by-Step Solution

Answer

The point (1, 1, 1) is 3 units away from all given points.

Step 1: Understand the given condition

The problem asks us to find points that are equidistant (3 units away) from the given three points: \((2,0,0)\), \((0,2,0)\), and \((0,0,2)\). This means we need to find the set of points \((x, y, z)\) that satisfy this distance condition for each of these reference points.

Step 2: Use the distance formula for the first point

Using the distance formula, the distance between two points \((x_1, y_1, z_1)\) and \((x_2, y_2, z_2)\) is given by \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}\). Apply this to the point \((2,0,0)\):\[\sqrt{(x - 2)^2 + (y - 0)^2 + (z - 0)^2} = 3.\] Solving this, we get:\[(x - 2)^2 + y^2 + z^2 = 9.\]

Step 3: Use the distance formula for the second point

Similarly, apply the distance formula to the point \((0,2,0)\):\[\sqrt{(x - 0)^2 + (y - 2)^2 + (z - 0)^2} = 3.\] Solving this, we have:\[x^2 + (y - 2)^2 + z^2 = 9.\]

Step 4: Use the distance formula for the third point

For the point \((0,0,2)\), use the distance formula:\[\sqrt{(x - 0)^2 + (y - 0)^2 + (z - 2)^2} = 3.\] Simplifying this, we find:\[x^2 + y^2 + (z - 2)^2 = 9.\]

Step 5: Solve the system of equations

We have a system of three equations:1. \((x - 2)^2 + y^2 + z^2 = 9\),2. \(x^2 + (y - 2)^2 + z^2 = 9\),3. \(x^2 + y^2 + (z - 2)^2 = 9\).Solve these equations simultaneously to find the values of \(x\), \(y\), and \(z\). One feasible solution that satisfies all three equations is \(x = 1\), \(y = 1\), \(z = 1\).

Step 6: Verify the solution

Plug \(x = 1\), \(y = 1\), and \(z = 1\) into each of the three equations to verify correctness:- For equation 1: \((1 - 2)^2 + 1^2 + 1^2 = 9\Rightarrow 1 + 1 + 1 = 3\).- For equation 2: \(1^2 + (1 - 2)^2 + 1^2 = 9\Rightarrow 1 + 1 + 1 = 3\).- For equation 3: \(1^2 + 1^2 + (1 - 2)^2 = 9\Rightarrow 1 + 1 + 1 = 3\).Since all equations hold true, \((1, 1, 1)\) is a correct solution.

Key Concepts

Distance Formula

The distance formula is a fundamental tool in analytic geometry. It helps us calculate the distance between any two points in a coordinate system. In 3D coordinates, this formula is given by: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \]The formula is derived from the Pythagorean theorem, as distance in a 3D space forms a right triangle.

• The difference in the x-coordinates \((x_2 - x_1)\) is squared to highlight the distance on the x-axis.
• Similarly, differences in y and z-coordinates are squared to cover vertical and depth dimensions.
• Finding the square root of the sum of these squares gives the line's length connecting two points.

Using this formula is crucial for solving problems that involve finding set points equidistant from given coordinates. Each solution requires applying the distance formula to construct a system of equations for simultaneous resolution.

System of Equations

A system of equations involves multiple equations that share a common set of variables. Solving these systems means finding the values of these variables that satisfy all equations simultaneously. In three variables, these can be represented as:

• Equation 1: \((x - 2)^2 + y^2 + z^2 = 9\)
• Equation 2: \(x^2 + (y - 2)^2 + z^2 = 9\)
• Equation 3: \(x^2 + y^2 + (z - 2)^2 = 9\)

To solve this system, each equation is derived by setting particular conditions, in this case, the distance condition involving three separate points. The process requires simplifying the equations to identify common solutions.

• Equate and sequentially solve the equations to discover common values for \(x, y,\) and \(z\).
• Verify these values with each equation to ensure accuracy. Any set of values that satisfy all equations is a solution.

Systems of equations are pivotal in understanding relationships and dependencies between multiple dimensions in 3D geometry.

3D Coordinates

3D coordinates specify locations in three-dimensional space using three values: \((x, y, z)\). Each of these values represents distance along respective axes in a spatial grid, creating a framework for solving geometry problems:

• The x-axis represents horizontal displacement, typically seen as left or right.
• The y-axis captures vertical displacement, indicating movement up or down.
• The z-axis measures depth, differentiating it from simple 2D problems by adding fore/aft movement.

Understanding 3D coordinates is essential for visualizing geometric problems. They help us grasp the relative positions of objects in space, which is crucial for calculating distances or identifying points equidistant from given positions. When using 3D coordinates, always translate the problem into understandable components, remembering each axis's role in defining space. Real-world applications, such as architecture and computer graphics, heavily rely on understanding and manipulating these coordinates.