/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 18 Find the distance between the gi... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Problem 18

Find the distance between the given two points. $$ (3,-1,-2) \text { and }(2,-3,-5) $$

Short Answer

Expert verified
The distance between the points is \(\sqrt{14}\).

Step by step solution

01

Understand the Problem

We are tasked with finding the distance between two points in a 3-dimensional space. The given points are \((3, -1, -2)\) and \((2, -3, -5)\).
02

Recall the Distance Formula

The distance formula for two points \((x_1, y_1, z_1)\) and \((x_2, y_2, z_2)\) in 3D space is given by:\[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \]
03

Subtract the Corresponding Coordinates

Calculate the differences of the corresponding coordinates:- \(x_2 - x_1 = 2 - 3 = -1\)- \(y_2 - y_1 = -3 - (-1) = -2\)- \(z_2 - z_1 = -5 - (-2) = -3\)
04

Square Each Difference

Calculate the squares of each of the differences:- \((-1)^2 = 1\)- \((-2)^2 = 4\)- \((-3)^2 = 9\)
05

Sum the Squares

Add the squared differences together:\[1 + 4 + 9 = 14\]
06

Take the Square Root

Find the distance by taking the square root of the sum:\[ \text{Distance} = \sqrt{14} \]
07

Simplify the Result

The distance remains as \(\sqrt{14}\) since 14 cannot further be simplified to an integer value.

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.

3-dimensional space
In our everyday world, we commonly navigate through three dimensions: length, width, and height. This is known as three-dimensional space. When we deal with 3D space in mathematics, we usually refer to it as the xyz-plane or xyz-coordinate system. Here, each point in space is defined by three coordinates.
  • The x-coordinate (abscissa) locates the position along the horizontal axis.
  • The y-coordinate (ordinate) locates the position along the vertical axis.
  • The z-coordinate locates the position along the depth axis, which is perpendicular to both x and y.
Understanding 3D space is crucial as it represents real-world objects more accurately than two dimensions, allowing for a comprehensive analysis involving volume and spatial relationships.
distance between points
To find the distance between two points in 3-dimensional space, we use the 3D distance formula. This formula extends the Pythagorean theorem (used in two dimensions) to accommodate an additional axis.
The formula is:\[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \]
Let's understand the process step by step:
  • Coordinate Differences: Start by finding the difference in each coordinate. This helps in understanding how far apart the points are along each axis individually. It's like measuring the length of a box between two corners.
  • Squares of Differences: Square these coordinate differences. Squaring ensures that the result is positive, and it elevates the concept to work in three dimensions.
  • Sum and Square Root: Add the squared differences and then take the square root. This aggregates the total 'straight-line' distance between the points, considering all dimensions together.
When you use these steps, you overcome the complexity of three dimensions and get an accurate representation of the distance.
coordinate geometry
Coordinate geometry, or Cartesian geometry, is the combination of algebra and geometry using a coordinate system. It allows for the precise representation and analysis of geometrical shapes by calculating their properties through equations and algebraic expressions.
In a 3-dimensional context, coordinate geometry helps us:
  • Find distances, as seen with our step-by-step problem.
  • Locate points precisely with three coordinates (x, y, z). Each point can be placed exactly in 3D space.
  • Understand the spatial relationship between various geometric shapes and planes.
Coordinate geometry is a powerful tool in various fields, from engineering to computer graphics, facilitating advancements by providing a method to model the complex physical world mathematically.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Find the double integral over the indicated region \(D\) in two ways. (a) Integrate first with respect to \(x\). (b) Integrate first with respect to \(y\). $$ \iint_{D} x e^{x^{2}} d A, D=\\{(x, y): 0 \leq x \leq 2,0 \leq y \leq 1\\} $$

Biology In measuring the volume of a rod-shaped bacterium (right circular cylinder), suppose an error of \(2 \%\) is made measuring its radius and a \(1 \%\) error is made in measuring its length. Using the tangent line approximation, estimate the greatest relative error \((|\Delta V| / V)\) in calculating its volume when \(r=1\) unit and \(h=10\) units.

Integrating first with respect to one of \(x\) or \(y\) is difficult, while integrating first with respect to the other is not. Find \(\iint_{D} f(x, y) d A\) the easier way. \(f(x, y)=1, D\) is the region in the first and fourth quadrants bounded by the curves \(x=1-y, y=x^{2}-1,\) and \(x=0\)

Suppose that a production function is given by \(P=f(x, y)\) and a cost function is given by \(C=p_{1} x+p_{2} y,\) where \(x\) is the number of items \(X\) made at a cost of \(p_{1}\) and \(y\) is the number of items \(Y\) made at a cost of \(p_{2}\). Use Lagrange multipliers to establish the important economic observation that if the firm wishes to produce \(P_{1}\) items while minimizing cost, then$$\frac{f_{x}}{p_{1}}=\frac{f_{y}}{p_{2}}$$

In Exercises 25 through \(28,\) determine whether the given function is increasing or decreasing at the point (a) (1,0) as \(y\) increases, (b) (0,1) as \(x\) increases, (c) (1,5) as \(y\) increases, and (d) (1,5) as \(x\) increases. $$ f(x, y)=x^{2}+y^{2}-x y-x-y $$
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Pure Maths

Read Explanation

Probability and Statistics

Read Explanation

Decision Maths

Read Explanation

Logic and Functions

Read Explanation

Mechanics Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.