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# Find the Cartesian equation for a plane parallel to a given vector and passing through the intersection of given planes

Let be the plane which is parallel to and which passes through the intersection of the planes

Find a Cartesian equation for .

First, we find the intersection of the two planes. This is the set of points which simultaneously satisfy

From the first equation we have

Plugging this into the second equation we have

This then gives us . So the line is the set of points where is arbitrary. Thus,

The plane must then contain this line and be parallel to the vector . Since it is parallel to its normal must be perpendicular to , thus,

and we must have

Therefore,

This has the Cartesian equation

# Prove that the intersection of two planes which are not parallel is a line

Prove that if and are two planes which are not parallel then they intersect in a line.

Proof. Let the Cartesian equations of and be given by

respectively. Then, the intersection is given by the common solutions of these two equations. Since and are not parallel, we know they do not have the same normal vector so that for all . Further, since the normals are nonzero, we know each equation has at least one nonzero coefficient. Without loss of generality, let . Then,

Substituting into the Cartesian equation for we have

is the set of solutions for the points on . But, we know at least one of or is nonzero, otherwise . Hence, we have the equation for a line. Therefore, is a line

# Prove a formula for the distance between a plane determined by three points and a point

1. If a plane is determined by the points prove that the distance from a point to this plane is given by

2. Using part (a) compute the distance in the case

1. Proof. We know the distance from a plane containing a point to a point not on the plane is given by the formula

Since the plane through is the set of points

we have . Thus, the distance from to is

2. For the given points we have

# Find a Cartesian equation for a plane parallel to a given plane and equidistant from a given point

Consider a plane given by the equation

Find the Cartesian equation for a plane parallel to this one and the same distance as this plane from the point .

Since the requested plane is parallel to the given plane we know that they must have the same normal vector, . Therefore, the Cartesian equation of the requested plane is of the form

From the previous exercise (Section 13.17, Exercise #19) we know the distance from to a plane is given by the formula

Therefore, the distance from the given plane to the point is

Since the distance from the point to the requested plane must be the same we must have

(Since the solution belongs to the other plane.)

# Prove some equations about distances between points and planes

1. Prove that the distance from the point to the plane

is given by the formula

2. Find the point on the plane which is nearest to the point .

1. Proof. By Theorem 13.6 (page 476 of Apostol) we know that the distance from a point to a plane is given by

2. A normal to the plane is given by . So, for any point . Further, the distance from to a point not on is minimal when where

Thus,

Naming to be the point we have

# Prove that the intersection of a line and plane which are not parallel contains exactly one point

Prove that the intersection of a line and a plane such that the line is not parallel to the plane contains one and only one point.

Proof. Denote the line by and the plane by . Let be the set of points

Since is not parallel to w know that its direction vector is not in the span of and . Further, by definition of a plane, we know the vectors and are linearly independent. Hence, are linearly independent. Then, any point in the intersection must be a solution to the system of equations

By the linear independence of we know this system has exactly one solution . Hence, contains exactly one point

# Find the parametric equation for a line through a point and parallel to two planes

We say that a line is parallel to a plane if the direction vector of the line is parallel to the plane. Let be the line containing the point and parallel to the planes

Find a vector parametric equation for .

The normal vectors of the planes are and . So, the direction vector of will be perpendicular to both of these,

From the first equation we have . Plugging this into the second equation we obtain , which then gives us . Since is arbitrary, we take to obtain a direction vector . Therefore, the vector parametric equation for the line is

# Prove that three planes with independent normal vectors intersect in exactly one point

Prove that three planes with normal vectors which are linearly independent intersect in exactly one point.

Proof. Let the normals of the three planes be given by

Then, the Cartesian equations of the three planes are given by

Since the normals are independent we know that they span the zero vector uniquely (by definition of independence). BY Theorem 12.7 (page 463 of Apostol) this implies that they span every vector in . Hence, the vector equation has a unique solution . Therefore, the system of equations has a unique solution. Hence, there is exactly one point on the intersection of three planes with linearly independent normal vectors

# Find all points on the intersection of three given planes

Consider three planes given by the Cartesian equations

Find all of the points which are on the intersections of these three planes.

The points on the intersection must satisfy the system of equations

To find the points satisfying these equations we row-reduce the coefficient matrix of the system (if you don’t have any linear algebra, then you can use Gaussian elimination the long way),

Therefore, we have , , and . So, the only point on the intersection is the point .

# Find the Cartesian equation of the plane parallel to given vectors with given intercept

Consider the plane which is parallel to both of the vector and and intersects the -axis at the point . Find the Cartesian equation of this plane.

Since contains the point and is parallel to and we have

Thus, the Cartesian equations for are

Hence,