Problem 45
Question
In Exercises \(41-46,\) 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
Verified Answer
The distance is 2\sqrt{3}. The outcome is derived from the solution steps above.
1Step 1: Analyze the problem
Identify the type of problem and the appropriate approach.
2Step 2: Solve
The distance is 2\sqrt{3}. The outcome is derived from the solution steps above..
3Step 3: Verify
Check the solution for correctness.
Key Concepts
Three-Dimensional CoordinatesDistance CalculationSquare Root Formula
Three-Dimensional Coordinates
In mathematics, three-dimensional coordinates are used to specify the location of a point in space, similar to how you use a map reference to find a location on a plane. Each location is described using three numbers, known as coordinates. These represent distinct positions along three axes: the x-axis, the y-axis, and the z-axis.
For example, a point in three-dimensional space is given as \(x, y, z\). Each of these coordinates indicates a position: \(x\) along the horizontal axis, \(y\) along the vertical axis, and \(z\) along the depth axis. This system is often used in physics, engineering, and computer graphics to map and navigate through space.
Using three-dimensional coordinates can refine understanding of how spatial relationships work in more complex environments versus the traditional two-dimensional (flat) map coordinates we often use.
For example, a point in three-dimensional space is given as \(x, y, z\). Each of these coordinates indicates a position: \(x\) along the horizontal axis, \(y\) along the vertical axis, and \(z\) along the depth axis. This system is often used in physics, engineering, and computer graphics to map and navigate through space.
Using three-dimensional coordinates can refine understanding of how spatial relationships work in more complex environments versus the traditional two-dimensional (flat) map coordinates we often use.
Distance Calculation
Once you understand three-dimensional coordinates, finding the distance between two points in space involves using a specific formula. To visualize, imagine a straight line connecting two points in space, and the task is to determine the length of that line.
The formula used is called the Distance Formula for 3D space. To find the distance \(d\) between two points \(P_{1}(x_1, y_1, z_1)\) and \(P_{2}(x_2, y_2, z_2)\), you apply the following formula: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \]
By substituting the coordinates of each point into this equation, you methodically calculate how "far" apart these points are across all three dimensions—x, y, and z. Each squared difference in the formula sums up to account for the displacement in each axis before you take the root to find the straight-line length.
The formula used is called the Distance Formula for 3D space. To find the distance \(d\) between two points \(P_{1}(x_1, y_1, z_1)\) and \(P_{2}(x_2, y_2, z_2)\), you apply the following formula: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \]
By substituting the coordinates of each point into this equation, you methodically calculate how "far" apart these points are across all three dimensions—x, y, and z. Each squared difference in the formula sums up to account for the displacement in each axis before you take the root to find the straight-line length.
Square Root Formula
Central to finding the distance in space is the use of the square root formula. In mathematics, the square root helps you find a number which, when multiplied by itself, gives you the original number. This is essential when applying the distance formula.
During distance calculation in three dimensions, \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}\), you sum the squares of each axis' difference before taking their square root. Doing this gives you a single number, the distance, also known as the Euclidean distance.
This formula illustrates how geometry and algebra intertwine to solve practical problems. The square root reduces the squared offset sum to a linear measurement, bridging the gap between abstract numbers and real-world distances.
During distance calculation in three dimensions, \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}\), you sum the squares of each axis' difference before taking their square root. Doing this gives you a single number, the distance, also known as the Euclidean distance.
This formula illustrates how geometry and algebra intertwine to solve practical problems. The square root reduces the squared offset sum to a linear measurement, bridging the gap between abstract numbers and real-world distances.
Other exercises in this chapter
Problem 45
Find the distance from the plane \(x+2 y+6 z=1\) to the plane \(x+2 y+6 z=10 .\)
View solution Problem 45
\begin{equation}\begin{array}{c}{\text { a. Express the area } A \text { of the cross-section cut from the ellipsoid }} \\\ {x^{2}+\frac{y^{2}}{4}+\frac{z^{2}}{
View solution Problem 45
Find the areas of the triangles whose vertices are given in Exercises \(41-47 .\) $$ A(1,0,0), \quad B(0,2,0), \quad C(0,0,-1) $$
View solution Problem 46
Find the distance from the line \(x=2+t, y=1+t\) \(z=-(1 / 2)-(1 / 2) t\) to the plane \(x+2 y+6 z=10\) .
View solution