### All Calculus 3 Resources

## Example Questions

### Example Question #71 : Normal Vectors

Find the normal vector to plane given by the equation of two vectors on the plane: and .

**Possible Answers:**

**Correct answer:**

To find the normal vector, you must take the cross product of the two vectors. Once you take the cross product, you get . In vector notation, this is .

### Example Question #72 : Normal Vectors

Calculate the norm of the vector:

**Possible Answers:**

None of the Above.

**Correct answer:**

Norm of the Vector is =

### Example Question #73 : Normal Vectors

Two vectors and are parallel to a plane. Find the normal vector to the plane.

**Possible Answers:**

**Correct answer:**

To find the normal vector to the plane, we must take the cross product of the two vectors. Using the 3x3 matrix , we perform the cross product.

Using the formula for the determinant of a 3x3 matrix

is

,

we get

### Example Question #74 : Normal Vectors

Find the normal vector of the plane that contains the lines and

**Possible Answers:**

**Correct answer:**

To find the normal vector to the plane, you must the the cross (determinant) between the vectors .The formula for the determinant of a 3x3 matrix is . Using the matrix in the problem statement, we get

### Example Question #75 : Normal Vectors

Find the normal vector to the plane given by the following vectors:

**Possible Answers:**

**Correct answer:**

The normal vector is given by the cross product of the vectors.

First, we must 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.

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:

### Example Question #76 : Normal Vectors

Find the normal vector to the plane that contains and

**Possible Answers:**

**Correct answer:**

The normal vector to the plane is found by taking the cross product of and . Using the formula for taking the cross product of two vectors, where and , we get . Using the vectors from the problem statement, we then get . In vector notation, this becomes .

### Example Question #77 : Normal Vectors

Find the normal vector to the plane containing the following vectors:

**Possible Answers:**

**Correct answer:**

The normal vector to the plane is given by the cross product of the two vectors in the plane.

First, 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.

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:

### Example Question #78 : Normal Vectors

Find the normal vector to the plane containing the following vectors:

**Possible Answers:**

**Correct answer:**

The normal vector to the plane is given by the cross product of the two vectors in the plane.

First, 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.

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:

### Example Question #79 : Normal Vectors

Find the normal vector to the plane given by the following vectors:

**Possible Answers:**

**Correct answer:**

The normal vector to the plane is given by the cross product of two vectors in the plane.

First, we must 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.

### Example Question #80 : Normal Vectors

Find the normal vector to the plane given the vectors in the plane

and

**Possible Answers:**

**Correct answer:**

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

First, we must 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.