# Calculus 3 : Normal Vectors

## Example Questions

### Example Question #81 : Normal Vectors

Find the normal vector of the plane that is parallel to the plane given by the equation

Explanation:

To solve the problem, we use the fact that two parallel planes have the same normal vector. The equation of on of the planes is given, and from that we know its normal vector is , which is the normal vector of the plane in question

### Example Question #81 : Normal Vectors

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

and

Explanation:

To find the normal vector to the plane containing vectors  and , we find the determinant of the 3x3 matrix

Plugging in the vectors and solving, we get

### Example Question #83 : Normal Vectors

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

and

Explanation:

To find the normal vector to the plane containing vectors  and , we find the determinant of the 3x3 matrix

Plugging in the vectors and solving, we get

### Example Question #84 : Normal Vectors

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

Explanation:

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:

The difference between zero and the zero vector is an important one, because the result of a cross product is always a vector (the dot product of two vectors gives a scalar).

### Example Question #85 : Normal Vectors

Find the normal vector to the plane containing the vectors  and

Explanation:

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

To find the cross product between the vectors  and , we find the determinant of the 3x3 matrix  which follows the formula

Applying to the vectors from the problem statement, we get

### Example Question #86 : Normal Vectors

Find the normal vector to the plane containing the vectors  and

Explanation:

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

To find the cross product between the vectors  and , we find the determinant of the 3x3 matrix  which follows the formula

Applying to the vectors from the problem statement, we get

### Example Question #87 : Normal Vectors

Find the normal vector to the following vectors:

Explanation:

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

Now, 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 #88 : Normal Vectors

Determine the vector normal to the plane created by the following two vectors:

Explanation:

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

So, we 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 #89 : Normal Vectors

Find the normal vector to the plane containing the vectors  and

Explanation:

To find the normal vector to a plane containing vectors  and , you take the cross product of the two vectors.

To find the cross product between the two vectors  and , you take the determinant of the 3x3 matrix

Using the vectors from the problem statement, we get

### Example Question #90 : Normal Vectors

Find the normal vector to the plane containing the vectors  and