### All Calculus 3 Resources

## Example Questions

### Example Question #61 : 3 Dimensional Space

Find the equation of the plane containing and 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 normal vector 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:

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

which simplifies to

### Example Question #62 : 3 Dimensional Space

Find the equation of the plane containing the point and the normal vector

**Possible Answers:**

**Correct answer:**

To find the equation of the plane that contains a point and has a normal vector , you use the formula:

Using the information from the problem statement, we get

Simplifying, we get

### Example Question #63 : 3 Dimensional Space

Find the equation of the plane containing the point and the normal vector

**Possible Answers:**

**Correct answer:**

To find the equation of the plane that contains a point and has a normal vector , you use the formula:

Using the information from the problem statement, we get

Simplifying, we get

### Example Question #64 : 3 Dimensional Space

Find the equation of the plane containing the point and the normal vector

**Possible Answers:**

**Correct answer:**

To find the equation of the plane that contains a point and has a normal vector , you use the formula:

Using the information from the problem statement, we get

Simplifying, we get

### Example Question #65 : 3 Dimensional Space

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

**Possible Answers:**

**Correct answer:**

To find the equation of any plane, we need a point on the plane and its normal vector. The normal vector of this plane is given by the equation of the parallel plane, due to the fact that they have the same normal vector. Using the point on the plane and the normal vector , we use the formula

Plugging in what we know, we get

Simplifying, we get

### Example Question #66 : 3 Dimensional Space

Find the equation of the plane from the points on the plane , and

Note: Use the point when forming the equation of the plane

**Possible Answers:**

**Correct answer:**

First, we need to form two vectors on the plane to get the normal vector to the plane. This is done from the following operation:

We then take the cross product of these vectors, which gets us the normal vector

Plugging the point and the normal vector into the equation of the plane, we get

Simplifying, we get

### Example Question #67 : 3 Dimensional Space

Find the equation of the plane tangent to the surface at the point where and .

**Possible Answers:**

**Correct answer:**

Find the equation of the plane tangent to the surface at the point where and .

** (1) **

The equation of the plane tangent to the surface defined by is given by the formula:

** (2)**

** **

In Equation (2) and are the partial derivatives of with respect to and , respectively. For this particular problem we have

Let's fill in Equation (2) term-by-term:

**Compute the partial derivative and then evaluate both at **

**- Partial with respect to x **

**- Partial with respect to **

Now fill in Equation (2) and simplify to get the equation of the tangent plane:

Therefore the equation of the tangent plane to the surface at the point is simply

### Example Question #68 : 3 Dimensional Space

Find the equation of the plane that contains the point and the normal vector

**Possible Answers:**

**Correct answer:**

To find the equation of the plane, we use the formula , where the point given is and the normal vector . Plugging in what we were given in the problem statement, we get . Manipulating the equation through algebra to isolate the variables, we get .

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