# Plane equation given two points and distance to parallel line

• queiroz
In summary, the equation 3b-2a+c= √[(-2b+2)²+(-c+2a-3)²+(b-1)²] has no solution and thus there is no unique plane that satisfies the given conditions.
queiroz

## Homework Statement

Find the plane through the points P = (1, 1, −1) and Q = (2, 1, 1) and parallel to a line r
{(1, 0, 2) + t(1, 0, 2), t ∈ R} with distance 1 to the plane

## Homework Equations

A=(a1,a2,a3),B(b1,b2,b3),A and B are vectors. AxB= det[i j k;a1 a2 a3;b1 b2 b3] ";" means change of line.

A.B= a1b1+a2b2+a3b3

vector projector of A in B is proj(B)A = (A . B /||B||² ) B

norm of vector projector of A in B is ||proj(B)A|| = ||(A . B /||B||² ) B||= (|A.B|/||B||²)||B||= |A.B|/||B||

## The Attempt at a Solution

I have the vector PQ=Q-P=(1,0,2) and named a point M(a,b,c) which gives me the vector PM=(a-1,b-1,c+1).
PQxPM will give me a vector perpendicular to the plane (scalar multiple of vector Normal N).
N= (-2b+2,-c+2a-3,b-1)
And the distance of the line to the plane is equal to the norm of the vector projection PP1 in the vector Normal when P1 is a point in the line and P in the plane. P1(1,0,2) and P(1,1,-1)
PP1=P1-P=(1,0,2)-(1,1,-1)=(0,-1,3)

||proj(N)PP1||= |PP1 . N|/||N||=1
{|(0,-1,3).(-2b+2,-c+2a-3,b-1)|/√[(-2b+2)²+(-c+2a-3)²+(b-1)²]}=1

And I get

3b-2a+c= √[(-2b+2)²+(-c+2a-3)²+(b-1)²] an equation with three unknown numbers.
I don't know how to proceed to find a,b and c.

Thanks

Last edited:
for your post! Your approach to solving this problem is correct so far. To find the values of a, b, and c, we can use the fact that the plane is parallel to the given line r. This means that the normal vector of the plane must be perpendicular to the direction vector of the line, which is (1, 0, 2). This gives us another equation:

N . (1, 0, 2) = 0
(-2b+2,-c+2a-3,b-1) . (1, 0, 2) = 0
-2b+4+2b-1=0
3=0

This is a contradiction, which means that there is no solution for a, b, and c. This makes sense since we are trying to find a plane that is both parallel to a line and has a distance of 1 to the plane. This is not possible, as a plane parallel to a line will either be the same plane or infinitely far away from it. Therefore, there is no unique solution for this problem.

## 1. How do you find the plane equation given two points and the distance to a parallel line?

To find the plane equation, first calculate the vector between the two given points. Then, use the cross product of this vector and the normal vector of the parallel line to find a third vector that lies in the plane. Finally, use the point-normal form of the plane equation to find the coefficients and solve for the equation.

## 2. What is the normal vector of a parallel line?

The normal vector of a parallel line is a vector that is perpendicular to the line. It can be found by taking the cross product of the direction vector of the line with any other vector that lies in the same plane.

## 3. How do you calculate the distance between a point and a parallel line?

To calculate the distance, first find the shortest distance between the given point and any point on the parallel line. This can be done by projecting the vector between the two points onto the normal vector of the line. Then, use the Pythagorean theorem to find the distance.

## 4. Can the plane equation be used to find the distance between a point and a non-parallel line?

No, the plane equation can only be used to find the distance between a point and a parallel line. For a non-parallel line, the distance can be found using the same method as in question 3, but the calculation will be slightly different.

## 5. What are some real-life applications of finding the plane equation using two points and the distance to a parallel line?

The plane equation can be used in various fields such as engineering, physics, and computer graphics. It can be used to calculate the orientation of an object, determine the intersection of two planes, or create 3D models of objects and environments. It is also commonly used in navigation systems to determine the distance between a plane and a point on the ground.

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