### All Calculus 3 Resources

## Example Questions

### Example Question #51 : 3 Dimensional Space

Find the equation of the plane given by a point on the plane and the normal vector

**Possible Answers:**

**Correct answer:**

To find the equation of a plane given a point on the plane and a normal vector to the plane , we use the following equation

Plugging in the information from the problem statement, we get

Rearranging, we get

### Example Question #52 : Equations Of Lines And Planes

Find the equation of the plane that contains the point and is parallel to the plane

**Possible Answers:**

**Correct answer:**

To find the equation of a plane that contains a point and a normal vector , we use the equation

Since we know the point on the plane as well as the normal vector (two parallel planes contain the same normal vector, so in this case it is , we can plug what we know into the equation

Rearranging, we get

### Example Question #51 : Equations Of Lines And Planes

Find the equation of the plane that contains the point and is parallel to the plane

**Possible Answers:**

**Correct answer:**

To find the equation of a plane that contains a point and a normal vector , we use the equation

Since we know the point on the plane as well as the normal vector (two parallel planes contain the same normal vector, so in this case it is , we can plug what we know into the equation

Rearranging, we get

### Example Question #54 : 3 Dimensional Space

Find the equation of the plane given by the following points:

, ,

**Possible Answers:**

**Correct answer:**

The equation of a plane is given by

where and is any point on the plane.

First, we must create two vectors out of the given points (by subtracting terminal and initial points):

,

Now, we can write the determinant in order to take the cross product of the two vectors, which will give us the normal vector:

where i, j, and k are the unit vectors corresponding to the x, y, and z direction respectively.

Next, we take the cross product. One can do this by multiplying across from the top left to the lower right, and continuing downward, and then subtracting the terms multiplied from top right to the bottom left:

Now that we have the normal vector, we can pick any point on the plane, and plug all of this into the formula above:

which simplified becomes

### Example Question #55 : 3 Dimensional Space

Determine the equation of the plane given by the following two vectors and the point :

**Possible Answers:**

**Correct answer:**

The equation of a plane is given by

where the normal vector is given by and a point on the plane denoted

To find the normal vector to the plane, we must take the cross product of the two vectors.

We must write the determinant in order to take the cross product:

where i, j, and k are the unit vectors corresponding to the x, y, and z direction respectively.

Next, we take the cross product. One can do this by multiplying across from the top left to the lower right, and continuing downward, and then subtracting the terms multiplied from top right to the bottom left:

Now that we have the normal vector and a point on the plane, we plug everything into the equation:

which simplifies to

### Example Question #56 : 3 Dimensional Space

Write the equation of the plane passing through and parallel to the lines and .

**Possible Answers:**

**Correct answer:**

The equation of the plane parallel to two lines is found by taking the point, and the normal vector to both lines (which in turn is normal - perpendicular - to the plane in question), and plugging them into the equation

where , and the point on the plane .

We must write the determinant in order to take the cross product of the two vectors (which gives us the normal vector):

where i, j, and k are the unit vectors corresponding to the x, y, and z direction respectively.

Next, we take the cross product. One can do this by multiplying across from the top left to the lower right, and continuing downward, and then subtracting the terms multiplied from top right to the bottom left:

We now have everything to plug into the equation.

We get

which simplified becomes

### Example Question #57 : 3 Dimensional Space

Find the equation of the plane that contains the points , , and

Use the point when forming the equation of the plane

**Possible Answers:**

**Correct answer:**

First, we need two vectors on the plane so we can form the normal vector to the plane. To do this, we take and , to form the vectors and .

Next we take the cross product of the vectors and

Using the formula for the cross product, we find that the normal vector is

We then use the formula for a plane with a normal vector and a point

Using the point , we get

Simplifying, we get

### Example Question #51 : Equations Of Lines And Planes

Find the equation of the plane given by the vectors parallel to the plane, and , and the point on the plane

**Possible Answers:**

**Correct answer:**

First, we need to find the normal vector to the plane, which is one by taking the cross product of the vectors that are parallel to the plane.

Using the formula for the cross product, we get that the normal vector is

Using the formula of a plane with a normal vector and containing a point , we get

Plugging in what we have, we get

Simplifying, we get

### Example Question #52 : Equations Of Lines And Planes

Find the equation of the plane parallel to the vectors , and containing the point

**Possible Answers:**

**Correct answer:**

If a plane is parallel to two vectors, then the vector perpendicular to those vectors must be perpendicular to the plane as well.

Using this fact, we take the cross product of the two vectors to get the normal vector, which is also the normal vector of the plane, .

The equation of a plane is given by

, where is a point on the plane.

So, we can write the determinant in order to take the cross product of the two vectors:

where i, j, and k are the unit vectors corresponding to the x, y, and z direction respectively.

Plugging this, and the point given, into our equation, we get

which simplifies to

### Example Question #60 : 3 Dimensional Space

Find the equation of the plane parallel to and and containing the point .

**Possible Answers:**

**Correct answer:**

The equation of a plane is given by

where the normal vector to the plane is and a point on the plane .

The plane is parallel to two vectors, so anything perpendicular to those vectors will be perpendicular to the plane.

So, we must take the cross product of the two vectors to get the vector normal to them (and therefore normal to the plane).

We can write the determinant in order to take the cross product of the two vectors:

where i, j, and k are the unit vectors corresponding to the x, y, and z direction respectively.

Plugging in all of our known information into the equation above, we get

which simplifies to