Find the Value of C in a Plane with Points (2,1,3) and (2,1,5) | Plane Equation

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In summary, the general equation for a plane is ax + by + cz = k, and it is satisfied by points (x,y,z). Given two specific points (2,1,3) and (2,1,5), the value of c can be found by solving the system of linear equations formed by plugging in the points into the equation. In this case, c = 0.
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FrostScYthe
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The general equation for a plane is ax + by + cz = k, where a, b, c, k are constants, and the plane is satisfied by points (x,y,z). If a specific plane contains both points (2, 1, 3) and (2, 1, 5), what is the value of c?

The answer is supposed to be c = 0, but I don't know how to get there.

I tried this [(2, 1, 3)-(2, 1, 5)]t + (2, 1, 5) so I get

x = 2
y = 1
z = 5 - 2t

but I don't know what to do from there
 
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  • #2
FrostScYthe said:
I tried this [(2, 1, 3)-(2, 1, 5)]t + (2, 1, 5) so I get

x = 2
y = 1
z = 5 - 2t
This is the parametric equation for a line, you want to describe a plane. Two points do not give a full description of a plane, so given two arbitrary points you would generally not be able to find 'c' in the equation. But in this particular case, you will see that if you plug in those two points to the original the equation things will cancel out and you'll be able to find 'c'.
 
  • #3
it's a system of linear equations:
you have:
2a+b+3c=k
2a+b+5c=k
solve it and get your answer.
 

FAQ: Find the Value of C in a Plane with Points (2,1,3) and (2,1,5) | Plane Equation

What is the formula for finding the value of C in a plane with two points?

The formula for finding the value of C in a plane with two points is: C = Ax + By + D, where A and B represent the x and y coefficients of the plane's normal vector, and D is the constant term.

How do you find the normal vector of a plane with two points?

To find the normal vector of a plane with two points, you can use the cross product of the two vectors formed by subtracting one point from the other. The resulting vector will have x and y coefficients for A and B, while the z coefficient will be -1. This vector can then be used to find the value of C.

Can the value of C be negative?

Yes, the value of C can be negative. This indicates that the plane is on the opposite side of the origin compared to the normal vector. A positive value of C would mean that the plane is on the same side as the normal vector.

What does the value of C represent?

The value of C represents the distance of the plane from the origin in the direction of the normal vector. It can also be thought of as the constant term in the equation of the plane.

Is there a different method for finding the value of C in a plane with more than two points?

Yes, with more than two points, you can use a system of equations to solve for the values of A, B, and C. This is done by setting up equations using the coordinates of the points and solving for the variables using techniques such as substitution or elimination.

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