# Calculus 3 : Equations of Lines and Planes

## Example Questions

### Example Question #141 : Calculus 3

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

Explanation:

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

### Example Question #142 : Calculus 3

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

Explanation:

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

### Example Question #143 : Calculus 3

Find the equation of the line that passes through the points  and

Note: The answer you obtain needs to be in vector form. Also, use the point  when forming the equation

Explanation:

To find the equation of the line, we need to find the vector that will be parallel to the line (its direction). This vector is formed by the points  and  (the ones from the problem statement.) We find the direction vector to be

We then pick a point on the line. We chose . We then form the equation of the line by using the formula

### Example Question #144 : Calculus 3

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

Explanation:

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

### Example Question #31 : 3 Dimensional Space

Find the equation of the plane containing the point , and is parallel to the plane with the equation

Explanation:

We were given a point on the plane, and we need the normal vector to the plane. It is known that two planes that are parallel to each other have the same normal vector, so in this case  (given by the equation of the other plane). To complete the problem, we use the equation , where  and the point on the plane is . Using the information we have, we get:

.  Through algebraic manipulation, we then get:

### Example Question #31 : Equations Of Lines And Planes

Find the equation of the plane containing the point , and is parallel to the plane with the equation

Explanation:

We were given a point on the plane, and we need the normal vector to the plane. It is known that two planes that are parallel to each other have the same normal vector, so in this case  (given by the equation of the other plane). To complete the problem, we use the equation , where  and the point on the plane is . Using the information we have, we get:

.  Through algebraic manipulation, we then get:

### Example Question #32 : Equations Of Lines And Planes

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

Explanation:

To solve the problem, we will use the formula for the equation of a plane with a normal vector  and a point :

We have the point, which from the problem statement is .

The normal vector is given by the equation of the plane that is in parallel to the one we are forming the equation for. That is .

Putting everything we know into the formula and solving, we get

Simplifying, we get

### Example Question #33 : Equations Of Lines And Planes

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

Explanation:

To solve the problem, we will use the formula for the equation of a plane with a normal vector  and a point :

We have the point, which from the problem statement is .

The normal vector is given by the equation of the plane that is in parallel to the one we are forming the equation for. That is .

Putting everything we know into the formula and solving, we get

Simplifying, we get

### Example Question #34 : Equations Of Lines And Planes

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

Explanation:

To find the equation of a plane, we use the normal vector  and a point on the plane . Using this information, we use the formula for a plane:

Using the information from the problem statement, we then get

Rearranging through algebra, we get:

### Example Question #35 : Equations Of Lines And Planes

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

Explanation:

To find the equation of a plane, we use the normal vector  and a point on the plane . Using this information, we use the formula for a plane:

To find the normal vector, it is known that two parallel planes have the same normal vector. Using this and the point on the plane, we then get

Rearranging through algebra, we get: