Proof, equation of a plan, linear algebra

In summary, the equation for a plane that contains the point P0 and has the normal vector N is N dot (P - P0) = 0. This can be proven by showing that if P is in the plane, then the dot product is 0, and if the dot product is 0, then P is in the plane.
  • #1
sciencegirl1
30
0

Homework Statement


Equation of a plan that contains the point p0=x0,y0,z0 and the normalvector N=A,B,C is N*P*P0=0 where P is(x,y,z)
=>A(x-xo)+B(y-y0)+c(z-z0)=0

Is it possible to prove this?
Can you help me if it is?
 
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  • #2
Maybe it should be N*(P-P0)=0 instead of N*P*P0?

If I understand correctly, you are supposed to prove that the plane that contains the point P0 and has normal N is given by the equation N*(P-P0)=0. This is true. Did you attempt to prove it?
 
  • #3
No I didn´t because I have no idea how to or if it´s possible.
Is it?
And if it is, can you tell me how to do it.
 
  • #4
P - P0 is a vector that is parallel to the plane, so this must be orhogonal to the normal vector. Therefore:

N dot (P - P0) = 0
 
  • #5
Hi sciencegirl1

drawing a picture always seems to help, but in line with yyat's comments, but the geometric reasoning is as follows

first take a point on the plane
[tex]\textbf{p}_0 = (x_0,y_0,z_0)[/tex]

next take any other point on the plan
[tex]\textbf{p} = (x,y,z)[/tex]

the difference is a vector parallel to the plane
[tex]\textbf{p}-\textbf{p}_0 = (x- x_0,y-y_0,z-z_0)[/tex]

so if we have a vector normal to the plane
[tex]\textbf{n}= (a,b,c)[/tex]

then by defintion this is perpindciular to any vector parallel to the plane, so
[tex]\textbf{n}*(\textbf{p}-\textbf{p}_0)= 0[/tex]

i'm not too sure what you want to do here, but if you want to prove this try writing a formula for a plane, find the normal vector, then take the dot product with the difference of any 2 points on the plane and see if you can show its 0
 
Last edited:
  • #6
If you have shown that if P is in the plane, then:

N dot (P-P0) = 0

Then what you need to show is that the reverse is also true. This is equivalent to saying that if P is not in the plane, you have

N dot (P - P0) not equal to zero.

This follows from the fact that if P - P0 is not in the plane, it must have a component along N.

So, you then have a necessary and sufficient condition for P to be in the plane: The equation

N dot (P - P0) = 0

This is therefore the equation of the plane.
 

1. What is a proof in mathematics?

A proof in mathematics is a logical argument that demonstrates the validity of a statement or theorem. It is a step-by-step process that shows how a conclusion can be logically derived from a set of assumed premises.

2. How do you write the equation of a plane in linear algebra?

The equation of a plane in linear algebra can be written in the form ax + by + cz = d, where a, b, and c are the coefficients of the variables x, y, and z, and d is a constant. This equation represents all the points in three-dimensional space that lie on the plane.

3. What is linear algebra used for?

Linear algebra is a branch of mathematics that deals with the study of linear equations and their representations in vector spaces. It is used in various fields such as computer science, physics, engineering, and economics to solve complex problems involving systems of linear equations and matrices.

4. How do you solve a system of linear equations using linear algebra?

To solve a system of linear equations using linear algebra, you can use techniques such as Gaussian elimination, Cramer's rule, or matrix inversion. These methods involve manipulating the coefficients of the equations and using properties of matrices to find the values of the variables that satisfy all the equations in the system.

5. What is the difference between a vector and a scalar in linear algebra?

In linear algebra, a vector is a quantity that has both magnitude and direction, and is represented geometrically as an arrow. A scalar, on the other hand, is a quantity that has only magnitude and does not have a specific direction. In other words, a scalar is a single number, while a vector is a combination of multiple numbers or variables.

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