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# Find a unit vector perpendicular to a given vector and parallel to a given plane

Find a unit length vector which is parallel to the plane with Cartesian equation and perpendicular to the vector .

The plane with Cartesian equation has intercepts . Therefore, the plane is the set of points

So, the vector being parallel to implies is in the span of . Therefore,

for some . Then, perpendicular to implies

Finally, since has unit length we have

This then gives us

Hence,

# Compute the volume of a tetrahedron with given vertices

Consider the tetrahedron with vertices at the origin and at the points where the plane

intersects the coordinate axes. Compute the volume of this tetrahedron.

First, the intercepts of the plane are given by . Then from a previous exercise (Section 13.14, Exercise #13) we know that the volume of a tetrahedron with vertices is

Letting we have

# Find a Cartesian equation for a plane through a point with normal vector making given angles

Let be the plane whose normal vector makes angles with the unit coordinate vectors and which contains the point . Find a Cartesian equation for the plane.

Since the normal vector to the plane makes angles with the unit coordinate vectors, we have

Hence, . So, the plane has a Cartesian equation of the form

Since it contains we have . Therefore, a Cartesian equation of the plane is

# Determine properties of a point whose movement in space is determined by a vector parametric equation

Consider a point moving in space with position at time given by

1. Prove that the motion of the point is along a line.
2. Find a vector parallel to this line.
3. Find the time at which the point intersects the plane with Cartesian equation .
4. What is the Cartesian equation for the plane parallel to the plane in part (c) which contains the point ?
5. Let be the plane perpendicular to containing the point . Find a Cartesian equation for .

1. Proof. We use the formula for the motion of the particle to compute

This is the parametric equation for the line through parallel to the vector

2. From part (a) we have a vector parallel to given by .
3. First, the line on which the point moves is the set of points

So, to find the intersection with the plane we compute

4. First, we have

Since we know the plane is parallel to the one in part (c) it has a Cartesian equation of the form

We compute . Hence, the plane has Cartesian equation

5. Since the plane is perpendicular to the line we know that it has a normal vector in the same direction as , so (from part (b)). Thus, we have a Cartesian equation of the form

Since the point

is on the plane we have . Therefore, the plane is given by

# 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