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Magazine Chapter 11: Problem 42
Find the distance from the point to the plane. $$(2,2,3), \quad 2 x+y+2 z=4$$
Short Answer
Expert verified
The distance is \(\frac{8}{3}\).
Step by step solution
01
Review the Formula for Distance from a Point to a Plane
The formula to find the distance, \( D \), from a point \((x_0, y_0, z_0)\) to a plane \(Ax + By + Cz + D = 0\) is:\[D = \frac{|Ax_0 + By_0 + Cz_0 + D|}{\sqrt{A^2 + B^2 + C^2}}\]First, we need to rewrite the given plane equation in the standard form.
02
Rewrite the Plane Equation in Standard Form
The given plane equation is \(2x + y + 2z = 4\). We rewrite it in the form \(Ax + By + Cz + D = 0\) by moving everything to one side:\[2x + y + 2z - 4 = 0\]Here, \(A = 2\), \(B = 1\), \(C = 2\), and \(D = -4\).
03
Identify the Point Coordinates
The coordinates of the given point \((x_0, y_0, z_0)\) are \((2, 2, 3)\).
04
Plug Values into the Distance Formula
Now, substitute the values into the distance formula:\[D = \frac{|2(2) + 1(2) + 2(3) - 4|}{\sqrt{2^2 + 1^2 + 2^2}}\]Calculate the numerator and the denominator separately.
05
Calculate the Numerator
The numerator is the absolute value of:\[|2 \cdot 2 + 1 \cdot 2 + 2 \cdot 3 - 4| = |4 + 2 + 6 - 4| = |8|\]Thus, the numerator is 8.
06
Calculate the Denominator
The denominator is:\[\sqrt{2^2 + 1^2 + 2^2} = \sqrt{4 + 1 + 4} = \sqrt{9} = 3\]
07
Compute the Distance
Substitute the calculated numerator and denominator back into the formula:\[D = \frac{8}{3}\]Thus, the distance from the point to the plane is \(\frac{8}{3}\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Distance Formula
The distance formula is a powerful mathematical tool used to determine the distance from a point to a plane in three-dimensional space. This concept is essential in coordinate geometry and is frequently applied in various fields, such as physics and engineering. For a point \((x_0, y_0, z_0)\) and a plane given by the equation \(Ax + By + Cz + D = 0\), the distance \(D\) from the point to the plane is calculated using the formula:
- \[D = \frac{|Ax_0 + By_0 + Cz_0 + D|}{\sqrt{A^2 + B^2 + C^2}}\]
- The absolute value in the numerator represents the perpendicular distance from the point to the plane, ensuring that the distance is always a non-negative value.
- The denominator, \(\sqrt{A^2 + B^2 + C^2}\), normalizes the equation by considering the orientation (or direction) of the plane.
Coordinate Geometry
Coordinate geometry, also called analytic geometry, is the study of geometry using a coordinate system. This system allows for the logical representation of geometric concepts using algebra.
In a three-dimensional context, where space is represented by the \((x, y, z)\) axes, coordinate geometry becomes a crucial tool for discovering relations among points, lines, and planes.
In a three-dimensional context, where space is represented by the \((x, y, z)\) axes, coordinate geometry becomes a crucial tool for discovering relations among points, lines, and planes.
- Points are defined by their coordinates, such as \((x_0, y_0, z_0)\) for a point in 3D space.
- A plane is represented by an equation, like \(Ax + By + Cz + D = 0\).
- Lines can be described by direction ratios and points through which they pass.
Plane Equation
The plane equation is a mathematical representation that describes a flat surface extending infinitely in two dimensions within a three-dimensional space. It is typically written in the general form: \(Ax + By + Cz + D = 0\).
In many mathematical problems, it is common to convert a plane equation to its standard form (where \(D = 0\)) to facilitate analysis and computation. Understanding the components of the plane equation is fundamental to solving distances and intersections in three-dimensional geometry.
- The coefficients \(A, B,\) and \(C\) correspond to the plane's normal vector, which is perpendicular to the plane itself.
- \(D\) is the constant term, which affects the position of the plane but not its orientation.
In many mathematical problems, it is common to convert a plane equation to its standard form (where \(D = 0\)) to facilitate analysis and computation. Understanding the components of the plane equation is fundamental to solving distances and intersections in three-dimensional geometry.
Three-Dimensional Geometry
Three-dimensional geometry, also known as solid geometry, involves the study of shapes and figures occupying three-dimensional space. Unlike two-dimensional geometry, it includes an additional dimension: depth.
By applying the concepts of vectors, planes, and points, three-dimensional geometry allows us to solve complex problems relating to distance, angles, and intersections in a space that more accurately reflects our physical environment.
- Points are located using three coordinates: \((x, y, z)\).
- Lines can be described by equations in vector form.
- Planes are represented by equations such as \(Ax + By + Cz + D = 0\).
By applying the concepts of vectors, planes, and points, three-dimensional geometry allows us to solve complex problems relating to distance, angles, and intersections in a space that more accurately reflects our physical environment.
