Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 11: Problem 26
Find the distance between the points. \((-2,3,2), \quad(2,-5,-2)\)
Short Answer
Expert verified
The distance between the points (-2,3,2) and (2,-5,-2) is \(4\sqrt{6}\)
Step by step solution
01
Write down the given points and the distance formula
The given points in the 3D space are A(-2,3,2) and B(2,-5,-2). The distance, d, between two points (x1, y1, z1) and (x2, y2, z2) in space follows this mathematical expression: \(d = \sqrt{(x2-x1)^2 + (y2-y1)^2 + (z2-z1)^2}\)
02
Substitute the values of the points in the formula
Substitute x1 = -2, y1 = 3, z1 = 2 and x2 = 2, y2 = -5, z2 = -2 into the distance formula. This gives \(d = \sqrt{(2-(-2))^2 + ((-5)-3)^2 + ((-2)-2)^2}\)
03
Simplify the expression
Simplify the equation to \(d = \sqrt{(4)^2 + (-8)^2 + (-4)^2} = \sqrt{16 + 64 + 16} = \sqrt{96}\)
04
Simplify the square root
96 is not a perfect square, but it can be simplified by finding the largest perfect square factor. The largest perfect square factor of 96 is 16, and \(96 = 16 \times 6\), so \(d = \sqrt{96} = \sqrt{16 \times 6} = 4\sqrt{6}\)
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
3D Coordinates
In a 3D space, each point is defined by three numerical values or coordinates. These coordinates represent positions along the three axes: the x-axis (horizontal), the y-axis (vertical), and the z-axis (depth). By combining these three axes, we can locate any point within the space. For example, a point could have coordinates \((-2, 3, 2)\), which correspond to:
- The x-coordinate \(-2\): How far left or right the point is located.
- The y-coordinate \(3\): How high up or down the point is.
- The z-coordinate \(2\): The depth of the point, how far forward or backward it is.
Distance Calculation
The problem of calculating the distance between two points in a 3D space requires a specific formula. This formula extends the Pythagorean theorem into three dimensions. It allows us to calculate the straight-line distance, or length of the hypotenuse, between two points. The formula is:\[d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2} \\]Here's how it works:
- Subtract the coordinates of the two points to find the difference along each axis.
- Square each of these differences to eliminate negative values and give them equal weight.
- Add these squared differences together.
- Finally, take the square root of this sum to determine the distance.
Euclidean Distance
Euclidean distance is a measure used to calculate the 'as-the-crow-flies' distance between points in a space, typically describing the shortest distance between them. It is commonly used in mathematics, physics, and various fields of computer science.
The Euclidean distance in 3D shares a similar process to 2D but involves an added dimension. In our example, the distance between points \((-2,3,2)\) and \(2,-5,-2)\) was calculated as follows:
The Euclidean distance in 3D shares a similar process to 2D but involves an added dimension. In our example, the distance between points \((-2,3,2)\) and \(2,-5,-2)\) was calculated as follows:
- Subtract the corresponding coordinates, then square the results: \(4^2, -8^2, -4^2\).
- Sum these squared numbers: \((16 + 64 + 16 = 96)\).
- Calculate the square root of this total to find the Euclidean distance: \(\sqrt{96} = 4\sqrt{6}\).
Problem-Solving Steps
Solving a problem like calculating Euclidean distance involves a series of logical steps. It's important to approach it systematically:
- **Identify Data:** Begin by listing the coordinates of the points involved in the calculation. Define each component clearly, like \(x_1, y_1, z_1\) and \(x_2, y_2, z_2\).
- **Apply Formula:** Substitute the identified values into the distance formula. This step directly links the problem to the equation used for calculation.
- **Simplify:** Break down each part of the equation, squaring differences first, then summing and simplifying toward a single square root expression.
- **Verify:** Double-check the results and simplification. Reassess your values to ensure no calculation errors occurred.
