Implicit Equatio for a Plane

In summary, the equation of the plane passing through the point (-2, 5, -5) and perpendicular to the line L(t) = <5+2t, 3, 4> is 2(x+2) + 0(y-5) + 0(z+5) = 0.
  • #1
shards5
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Homework Statement


An implicit equation for the plane passing through the point (-2,5,-5) that is perpendicular to the line L(t) = <5+2t,3,4> is ...?

Homework Equations



a(x-x0) + b(y-y0) + c(z-z0) = 0

The Attempt at a Solution


So in order to find the equation of the plane I would need a normal vector and I got that using the following steps:
Find a point on the line: L(1) = <7, 3, 4>
Get a vector from the given point to the newly found point. (7,3,4) - (-2,5,-5) = (9,-2,9)
Find the normal direction of the line: <2,0,0>
Use the cross product to find the normal vector. (9,-2,9) x <2,0,0> = <0,18,4>
Therefore I have the equation: 18(y-5) + 4(z+5) = 0, but this is wrong. What am I doing wrong?
 
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  • #2
shards5 said:

Homework Statement


An implicit equation for the plane passing through the point (-2,5,-5) that is perpendicular to the line L(t) = <5+2t,3,4> is ...?

Homework Equations



a(x-x0) + b(y-y0) + c(z-z0) = 0

The Attempt at a Solution


So in order to find the equation of the plane I would need a normal vector and I got that using the following steps:
Find a point on the line: L(1) = <7, 3, 4>
Get a vector from the given point to the newly found point. (7,3,4) - (-2,5,-5) = (9,-2,9)
Find the normal direction of the line: <2,0,0>
Use the cross product to find the normal vector. (9,-2,9) x <2,0,0> = <0,18,4>
Therefore I have the equation: 18(y-5) + 4(z+5) = 0, but this is wrong. What am I doing wrong?

Your normal is correct, <2, 0, 0>, which you can read directly from the equation of the line. The numbers in the normal are the coefficients of t, which are 2, 0, and 0.

Once you have a normal <A, B, C> to the plane, and you know a point (a, b, c) on the plane, the equation of the plane is A(x -a) + B(y - b) + C(z - c) = 0.

Calculating the cross product is a waste of time, since you already have a normal to the plane, namely <2, 0, 0>.
 

1. What is an implicit equation for a plane?

An implicit equation for a plane is a mathematical representation of a plane in three-dimensional space that does not explicitly state the variables x, y, and z. Instead, it is written in the form of a polynomial equation, with terms containing x, y, and z, as well as constants. This equation can be used to describe the points on the plane and is useful in solving problems involving planes in geometry and physics.

2. How is an implicit equation for a plane different from an explicit equation?

An explicit equation for a plane explicitly states the variables x, y, and z in terms of each other, such as z = 3x + 2y. On the other hand, an implicit equation does not explicitly state the variables and is usually in the form of a polynomial equation, such as 2x + 3y + 4z = 10. Implicit equations are often more complex and can describe a wider range of planes, including those that are not parallel to any of the coordinate axes.

3. How can I graph an implicit equation for a plane?

To graph an implicit equation for a plane, you can use a graphing calculator or plotting software. First, solve the equation for one variable, such as z, in terms of the other two variables, x and y. Then, input the equation into the graphing calculator or software and adjust the viewing window to see the plane. You can also plot points on the plane by choosing values for x and y and solving for z, then connecting the points to create a visual representation of the plane.

4. What are some real-world applications of implicit equations for planes?

Implicit equations for planes have many real-world applications, including in engineering, physics, and computer graphics. Engineers use them to model the surfaces of objects and structures, while physicists use them to describe the motion of objects in space. In computer graphics, implicit equations for planes are used to create three-dimensional shapes and animations. They are also used in navigation and mapping systems, as well as in aircraft and spacecraft design.

5. How do implicit equations for planes relate to vectors and normal vectors?

Implicit equations for planes are closely related to vectors and normal vectors. The coefficients of x, y, and z in the equation represent the components of the normal vector to the plane, which is a vector that is perpendicular to the plane's surface. This normal vector is important for calculating the angle between two planes, determining the direction of motion of an object on the plane, and finding the equation of a line that is perpendicular to the plane. Vectors can also be used to translate and rotate the plane in space.

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