Equation of Plane: Finding a Normal Vector for Three Given Points

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In summary, the task is to find an equation of a plane containing three given points, with the coefficient of x being 9. This means that the x coordinate of the normal vector will also be 9, and the equation can be adjusted to have a coefficient of 9 by multiplying both sides by a constant.
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Homework Statement


Find an equation of a plane containing the three points (1, -1, 0), (5, 4, 1), (5, 5, 3) in which the coefficient of x is 9. What does it mean when it says the coefficient of x is 9?


Homework Equations


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


The Attempt at a Solution


Since the equation of the plane needs a normal vector I can get the normal vector using the following steps:
ab = (5, 4, 1) - (1, -1, 0) = (4,5,1)
ac = (5, 5, 3) - (1, -1, 0) = (4,6,3)
Normal Vector = (4,5,1) x (4,6,3) = <9,-8,4>
So I would get 9(x-1) - 8(y+1) +4(z-0) = 0 but where does the coefficient of x is 9 come in?
 
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shards5 said:

Homework Statement


Find an equation of a plane containing the three points (1, -1, 0), (5, 4, 1), (5, 5, 3) in which the coefficient of x is 9. What does it mean when it says the coefficient of x is 9?


Homework Equations


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


The Attempt at a Solution


Since the equation of the plane needs a normal vector I can get the normal vector using the following steps:
ab = (5, 4, 1) - (1, -1, 0) = (4,5,1)
ac = (5, 5, 3) - (1, -1, 0) = (4,6,3)
Normal Vector = (4,5,1) x (4,6,3) = <9,-8,4>
So I would get 9(x-1) - 8(y+1) +4(z-0) = 0 but where does the coefficient of x is 9 come in?
The coefficient of x in your plane equation is 9 (9x - 9 - 8y - 8 + 4z = 0). As it turned out, 9 was the x coordinate of the normal.

If you had come up with a normal of, say, <3, -8/3, 4/3>, your equation would have been 3(x - 1) - 8/3(y + 1) + 4/3(z - 0) = 0. You could have adjusted the equation to get an x coefficient of 9 by multiplying both sides of the equation by 3.

I guess that's where they were going with this problem...
 

1. What is the equation of a plane?

The equation of a plane is a mathematical expression that describes the relationship between the coordinates of points in a three-dimensional space. It is typically written in the form Ax + By + Cz + D = 0, where A, B, and C are the coefficients of the variables x, y, and z, and D is a constant term.

2. What are the variables in the equation of a plane?

The variables in the equation of a plane are x, y, and z, which represent the coordinates of a point in three-dimensional space. These variables are used to describe the position of a point on the plane.

3. How is the equation of a plane different from the equation of a line?

The equation of a plane is different from the equation of a line because it describes a two-dimensional surface in three-dimensional space, while the equation of a line describes a one-dimensional object in two-dimensional space. Additionally, the equation of a plane has three variables (x, y, and z), while the equation of a line has two variables (x and y).

4. What information can be determined from the equation of a plane?

The equation of a plane can provide information about the orientation and position of the plane in three-dimensional space. It can also be used to determine the distance of a point from the plane, as well as the intersection points between two planes.

5. How is the equation of a plane used in real-world applications?

The equation of a plane is used in various fields such as engineering, physics, and computer graphics to model and analyze three-dimensional objects and systems. It is also used in navigation and aviation to calculate the trajectory of objects in three-dimensional space.

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