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Magazine Chapter 2: Problem 286
Find the general equation of the plane that passes through the origin and is perpendicular to the line of intersection of planes \(-x+y+2=0\) and \(z-3=0\)
Short Answer
Expert verified
The general equation of the plane is \(x + y = 0\).
Step by step solution
01
Identify the Intersection Line
The line of intersection of two planes can be found by identifying a vector that lies in the direction of the intersection. The planes given are \(-x+y+2=0\) and \(z-3=0\). The direction vector of the line of intersection can be found using the cross product of the normal vectors of these two planes.
02
Find Normal Vectors of the Two Planes
The normal vector of the plane \(-x+y+2=0\) is \(\mathbf{n_1} = [-1, 1, 0]\), and the normal vector of the plane \(z-3=0\) is \(\mathbf{n_2} = [0, 0, 1]\).
03
Calculate the Direction Vector
The direction vector of the line of intersection is given by the cross product \(\mathbf{d} = \mathbf{n_1} \times \mathbf{n_2}\). Calculate it:\[\mathbf{d} = [-1, 1, 0] \times [0, 0, 1] = [1, 1, 0]\]
04
Determine the Normal Vector of the Plane
The plane we're looking for is perpendicular to the line of intersection, so it must have a normal vector that is the same as the direction vector of the line of intersection. Therefore, the normal vector of the plane is \(\mathbf{d} = [1, 1, 0]\).
05
Write the General Equation of the Plane
Since the plane passes through the origin, this means that the equation is in the form \(ax + by + cz = 0\). With a normal vector \(\mathbf{d} = [1, 1, 0]\), the general equation is:\[1\cdot x + 1\cdot y + 0\cdot z = 0\]Simplifying, the equation of the plane is \(x + y = 0\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Intersection of Planes
Understanding the intersection of two planes is a key concept in geometry. When two planes intersect, they form a line. This line is described by a unique direction vector that represents the continuous direction in which the intersection extends. To find this line, you need to determine the points that satisfy the equations of both planes simultaneously.
Sometimes, it's easier to find a common line by using normal vectors. These normal vectors are perpendicular to the planes and can guide us to finding the intersection line. In the case of the planes given by \(-x+y+2=0\) and \(z-3=0\), we use the normal vectors of these planes to calculate the direction of their intersection.
Sometimes, it's easier to find a common line by using normal vectors. These normal vectors are perpendicular to the planes and can guide us to finding the intersection line. In the case of the planes given by \(-x+y+2=0\) and \(z-3=0\), we use the normal vectors of these planes to calculate the direction of their intersection.
Normal Vector
A normal vector is essential in understanding planes. It is a vector that is perpendicular, or normal, to the surface of a plane. The normal vector provides significant information about the orientation of the plane in three-dimensional space.For the plane \(-x+y+2=0\), the normal vector is \([-1, 1, 0]\).
This means any line with this direction is perpendicular to the plane. Similarly, the plane \(z-3=0\) has the normal vector \([0, 0, 1]\), showing it stretches along the x and y axes. By using these normal vectors, we can solve various geometry problems, like finding the line of intersection between planes, by computing their cross product.
This means any line with this direction is perpendicular to the plane. Similarly, the plane \(z-3=0\) has the normal vector \([0, 0, 1]\), showing it stretches along the x and y axes. By using these normal vectors, we can solve various geometry problems, like finding the line of intersection between planes, by computing their cross product.
Cross Product
The cross product is a vector operation used to find a vector that is perpendicular to two given vectors, highly useful in three-dimensional vector calculations.To find the line of intersection of two planes, the cross product (\(\times\)) of their normal vectors gives a direction vector. For example, the two normal vectors \([-1, 1, 0]\) and \([0, 0, 1]\) from the given planes intersect to form a new vector via their cross product:\[[-1, 1, 0] \times [0, 0, 1] = [1, 1, 0]\]This result, \([1, 1, 0]\), represents the direction vector of the line where the two planes intersect. The cross product operation is pivotal for visualizing and defining spatial relationships.
Direction Vector
The direction vector is a vector that gives us insight into the direction in which a line runs. In geometry, it helps define the characteristics of lines, particularly in space where precise direction is crucial.In the case of plane intersections, once we have the direction vector from the cross product, \([1, 1, 0]\), it becomes invaluable. This direction vector describes the line of intersection between the two planes. It is not only parallel but also indicative of how the intersection line extends in space. Understanding the direction vector empowers you to write parametric equations of lines and understand spatial alignments.
