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Chapter 11: Problem 32

In Exercises 27-36, find the distance between the points. \((1, 1, -7), \quad (-2, -3, -7)\)

Short Answer

Expert verified
5

Step by step solution

01

Identify the coordinates

First, recognize that the given points are in the form \((x_1, y_1, z_1)\) and \((x_2, y_2, z_2)\). From the given data, we have \((x_1, y_1, z_1) = (1, 1, -7)\) and \((x_2, y_2, z_2) = (-2, -3, -7)\).
02

Apply the distance formula

The distance formula in three dimensional space is \(d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2+(z_2-z_1)^2}\). When you replace the points' coordinates into the formula, you will obtain \(d = \sqrt{(-2-1)^2 + (-3-1)^2 + (-7-(-7))^2}\)
03

Simplify the expression

Now simplify each term inside the square root. You will get \(d = \sqrt{(-3)^2 + (-4)^2 + 0}\).
04

Calculate the distance

Simplify the equation further you will get \(d = \sqrt{9 + 16 + 0} = \sqrt{25}\)
05

Final step

The square root of 25 is 5. Therefore, the distance between the two points is 5 units.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

3D Geometry
3D geometry involves the study of objects in a three-dimensional space, which includes the added complexity of depth along with height and width. This type of geometry is fundamental for understanding structures and shapes that have volume.
  • Dimensions: In 3D geometry, you have three coordinates: x (width), y (height), and z (depth). This is what distinguishes 3D geometry from its 2D counterpart which only utilizes x and y coordinates.
  • Applications: 3D geometry is commonly used in various fields including architecture, engineering, and computer graphics.
Understanding 3D geometry helps in visualizing objects in a more realistic way. Each point in a 3D space is determined by its position relative to these three coordinates.
Distance Calculation
Calculating the distance between two points in a 3D space involves extending the Pythagorean theorem into three dimensions. This is key to ensuring accurate measurements across complex spatial relationships.
Let's briefly go through the distance formula step-by-step:
  • Identify Coordinates: Find the coordinates of the two points. For example, if you have two points like \((x_1, y_1, z_1)\) and \((x_2, y_2, z_2)\).
  • Apply the Formula: Use the 3D distance formula: \[d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2}\]
  • Substitute Values: Insert the coordinates into the formula and calculate the squared differences for each dimension.
  • Solve: Simplify the expression to find the distance.
This method ensures that you measure the straight-line distance or the 'crow flies' distance between the two points in space.
Coordinate System
A coordinate system is a system that uses numbers (coordinates) to uniquely determine the position of a point or a geometric element on a plane or in space. The coordinate system is central to the understanding of both 2D and 3D geometry.
  • Cartesian Coordinates: One of the most widely used systems, where each point in space is defined by its x, y, and z coordinates.
  • Reference Point: The origin, usually \((0, 0, 0)\), serves as a reference point for determining the position of other points.
  • Axes: The three axes, x, y, and z, are usually perpendicular to each other, forming a three-dimensional grid.
Coordinate systems help in visualizing and calculating the positions and relationships between objects in space. Understanding how to work with coordinates allows for precise modeling, planning, and construction in all three dimensions.

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Most popular questions from this chapter

In Exercises 55-58, use the triple scalar product to find the volume of the parallelepiped having adjacent edges \(\textbf{u}, \textbf{v},\) and \(\textbf{w}\). \(\textbf{u}=\textbf{i}+\textbf{j}\) \(\textbf{v}=\textbf{j}+\textbf{k}\) \(\textbf{w}=\textbf{i}+\textbf{k}\)

In Exercises 5-10, find a set of (a) parametric equations and (b) symmetric equations for the line through the point and parallel to the specified vector or line. (For each line, write the direction numbers as integers.) \(\quad \quad \textit{Point} \quad \quad \quad \quad \quad \quad \textit{Parallel to}\) \(\quad (0, 0, 0) \quad \quad \quad \quad \quad \textbf{v} = \langle 1, 2, 3 \rangle\)

In Exercises 37-40, determine whether the planes are parallel, orthogonal, or neither. If they are neither parallel nor orthogonal, find the angle of intersection. \(5x-3y+z=4\) \(x+4y+7z=1\)

In Exercises 27-30, find the general form of the equation of the plane passing through the three points. \((0, 0, 0), (1, 2, 3), (-2, 3, 3)\)

In Exercises 31-36, find a unit vector orthogonal to \(\textbf{u}\) and \(\textbf{v}\). \(\textbf{u} = \textbf{i}+2\textbf{j}\) \(\textbf{v} = \textbf{i}-3\textbf{k}\)
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