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# 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,

# 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

# Determine if two given planes are parallel and find points on their intersection

Let and let . Let be the plane determined by the Cartesian equation .

1. Determine if the two planes are parallel.
2. If we define to be the plane with Cartesian equation find two points on .

1. We know for any point , there is a unique plane parallel to containing . We pick a point on that is not on and show that the unique plane parallel to containing is, in fact, .
The point is on since

It is not on since

has no solution . So, the unique plane parallel to containing is

Then, we obtain the Cartesian equation of this plane. We have

This gives us . Then, which implies

Hence,

Hence, this has the Cartesian equation . But this is the plane ; hence, is parallel to .

2. The points and are both in the intersection since they both satisfy the equations

# Determine which points are on the plane with Cartesian equation 3x – 5y + z = 9

Let be the plane with Cartesian equation .

1. Determine which of the points are on the plane.
2. Find vectors such that .

1. The point since

The point since

The point since

2. We know . So, let

Since let

Then we have

# Determine a Cartesian equation for given planes

For each of the following planes, find a linear Cartesian equation of the form

that describes the plane.

1. The plane through spanned by .
2. The plane through the points .
3. The plane through the point parallel to the plan through spanned by and .

1. The plane through spanned by and is the set of points

Therefore, we have the parametric equations

Then we want to solve for in terms of . From the first equation we have

From the second equation we then have

Which gives us

So, from the third equation we then have

Thus,

is the requested linear Cartesian equation.

2. The plane through the three points is the set of points

But, and are in the linear span of since

Thus, this plane is equal to the plane in part (a). Hence, we have the linear Cartesian equation,

3. Again, this is the same plane as in parts (a) and (b) since the span of and is the same as the span of and . Hence, the requested linear Cartesian equation is