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# Find a vector parametric equation for a line containing a given point and perpendicular to a given plane

Let be the line which contains the point and is perpendicular to the plane given by the equation . Find a vector parametric equation for .

From the Cartesian equation for the plane we have is a normal vector. So, is the line through which is parallel to . Thus, the vector parametric equation for the line is

# Find the Cartesian equation of plane through a given point and with a given perpendicular line

We say that a line parallel to a vector (non-zero) is perpendicular to a plane if is normal to . Given that a plane goes through the point and that the line through the points and is perpendicular to find the Cartesian equation of .

First, . Therefore, the Cartesian equation of is of the form

Since is on the plane we have . Thus, the Cartesian equation of is

# Determine the angle between planes with given Cartesian equations

Consider two planes with Cartesian equations,

Determine the angle between the planes.

For the two planes we have normals and . Therefore, the angle between the planes is

# Find the Cartesian equation of a plane given three points on the plane

Let , , and be three points on a plane. Find the Cartesian equation for the plane.

First, letting , , and we compute a normal to the plane

So the Cartesian equation of the plane is of the form

Since is on the plane we have . Hence, the Cartesian equation for the plane is

# Find properties of a plane given three points that determine it

Let be the plane determined by the three points , , and . Find the following:

1. A normal vector to the plane.
2. A Cartesian equation for the plane.
3. The distance between the plane and the origin.

1. Denote the points by , and , then we can compute a normal vector by

Therefore, is a normal vector to the plane.

2. Since is normal to the plane we have a Cartesian equation of the form

Then, since is on the plane we have . Hence, the Cartesian equation is

3. The distance from the origin is

# Establish properties of four planes with given Cartesian equations

Consider four planes with the Cartesian equations:

1. Establish that two of them are parallel and the other two are perpendicular.
2. For the two parallel planes, find the distance between them.

1. The second and fourth planes are parallel since they have the same normal vector, .
To see that the first and third are perpendicular, we denote the normal vectors by and , respectively, and compute

Hence, they are perpendicular.

2. Denoting the second and fourth planes by and , respectively we have Cartesian equations

Therefore, the distance between them is

# Find the Cartesian equation of a plane passing through a point and parallel to a given plane

Let be a plane which passes through the point and is parallel to the plane given by the equation . Find the Cartesian equation of . Further, find the distance between the two planes.

Let denote the plane given by the equation . Since and are parallel, we know they share a common normal vector. Therefore, the Cartesian equation of is of the form

Since is on , we have

Hence, the Cartesian equation of is

The distance between and is then

# Give a Cartesian for planes through given points spanned by given vectors

Consider the vectors

1. Find a nonzero vector perpendicular to both and .
2. Find a Cartesian equation for the plane through which is spanned by and .
3. Find a Cartesian equation for the plane through which is spanned by and .

1. Since and are independent, we can take

2. From part (a) we have is perpendicular to both and , so a Cartesian equation for the plane is given by

Further, since the point is on the plane, we must have . Hence, the Cartesian equation for the plane is

3. Again, we have a Cartesian equation for the plane given by

Since is on the plane we must have

Hence, the Cartesian equation for the plane is given by

# Prove that there is exactly one plane through a line and a point not on the line

Let be a line and a point not on . Prove that there is exactly one plane through containing every point of .

Proof. Existence follows from a previous exercise (Section 13.8, Exercise #12) by letting be any two distinct points on , then there is a plane containing and every point on the line through and .
Now, to prove there is only one such plane we apply Theorem 13.10 to the points to conclude the plane that contains is unique. But, if we choose any other two points on , the plane must still contain and (since it contains every point on ); hence, it is the same plane . Thus, there is exactly one plane containing and

# Find a Cartesian equation for a plane through a point containing a given line

Let be the line through the point and parallel to the vector and let be a point not on . Find a Cartesian equation for the plane which passes through and entirely contains .

The line is the set of points

Then, the plane is the set of points

Then, to get the Cartesian equation, we have

The first two equations give and so . This implies , and is arbitrary. So, the plane is described by the equation .