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Chapter 11: Problem 41

Find the distance between points \(P_{1}\) and \(P_{2}\). $$P_{1}(1,1,1), \quad P_{2}(3,3,0)$$

Short Answer

Expert verified
The distance between the points is 3 units.

Step by step solution

01

Identify the Coordinates

The first step is to clearly identify the coordinates of points \(P_1\) and \(P_2\). The coordinates of \(P_1\) are \((1, 1, 1)\) and the coordinates of \(P_2\) are \((3, 3, 0)\).
02

Apply the Distance Formula in 3D

To find the distance between two points in three dimensions, we use the distance formula: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \] where \((x_1, y_1, z_1)\) are the coordinates of \(P_1\) and \((x_2, y_2, z_2)\) are the coordinates of \(P_2\).
03

Calculate the Differences

Substitute the coordinates of the points into the formula: \((3, 3, 0)\) and \((1, 1, 1)\). First, calculate each component's difference: \((3 - 1), (3 - 1), (0 - 1)\) which are \(2, 2, -1\) respectively.
04

Substitute and Simplify

Now substitute these differences into the distance formula: \[ d = \sqrt{(2)^2 + (2)^2 + (-1)^2} \] Simplify each term: \(d = \sqrt{4 + 4 + 1}\).
05

Calculate the Final Answer

Add up the values inside the square root: \(d = \sqrt{9}\). Take the square root to find the distance: \(d = 3\).

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Key Concepts

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

Coordinates
Coordinates are numerical values that define the position of a point in space. In a three-dimensional space, every point is described using three numbers, often represented as
  • x
  • y
  • z
These values are known as the coordinates of the point. For instance, the coordinates
  • (1, 1, 1) represents the point \(P_1\)
  • (3, 3, 0) represents the point \(P_2\)
Each of these numbers tells us the position of the point along respective axes in the three-dimensional coordinate system.
The x coordinate represents the horizontal distance from the origin,
the y coordinate represents the vertical magnitude,
and the z coordinate represents the depth.
Three-Dimensional Space
Three-dimensional space is a geometric setting characterized by three perpendicular axes:
  • x-axis
  • y-axis
  • z-axis

This space is the natural settings where the concept of coordinates system is applied to define the position of a point fully.
In the Cartesian coordinate system, the intersection of these axes at point (0,0,0) is called the origin. Each axis serves to measure distances in a separate direction, making it possible to describe the location precisely:
  • The x-axis measures horizontal distance.
  • The y-axis measures vertical distance.
  • The z-axis measures the forwards and backwards positions.
Every point in this space can be uniquely identified using its coordinates, allowing us to calculate distances and navigate through this dimensional environment.
Distance Calculation
Calculating distances between two points in three-dimensional space can be achieved using the Distance Formula. This formula helps find the straight-line distance between
  • two points with coordinates
    • \((x_1, y_1, z_1)\)
    • \((x_2, y_2, z_2)\)
The formula is written as:\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \]This mathematical representation is derived from the Pythagorean theorem, extended for three dimensions.
Here's how it works in practice:
  • Find the differences between the corresponding coordinates \((x, y, z)\).
  • Square each difference value to eliminate negative results and measure absolute deviation.
  • Sum these squared values.
  • Compute the square root of the resulting sum to find the distance \(d\).
This clear process allows for calculating the actual length of the line segment connecting two points in 3D space.

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Most popular questions from this chapter

Find parametrizations for the lines in which the planes. $$3 x-6 y-2 z=3, \quad 2 x+y-2 z=2$$

Find equations for the spheres whose centers and radii are given. Center \(\left(-1, \frac{1}{2},-\frac{2}{3}\right)\) Radius \(\frac{4}{9}\)

Sketch the surfaces in Exercises \(13-44\) $$16 x^{2}+4 y^{2}=1$$

Find the point of intersection of the lines \(x=t, y=-t+2,\) \(z=t+1,\) and \(x=2 s+2, y=s+3, z=5 s+6,\) and then find the plane determined by these lines.

Find the acute angles between the planes in Exercises \(49-52\) to the nearest hundredth of a radian. $$2 x+2 y+2 z=3, \quad 2 x-2 y-z=5$$
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