Calculus 3 : Equations of Lines and Planes

Study concepts, example questions & explanations for Calculus 3

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Example Questions

Example Question #41 : 3 Dimensional Space

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

Possible Answers:

Correct answer:

Explanation:

To find the equation of the plane with a normal vector  and a point , we use the formula

Using the information from the problem statement, we get 

This simplifies to 

 

Example Question #42 : 3 Dimensional Space

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

Possible Answers:

Correct answer:

Explanation:

To find the equation of the plane with a normal vector  and a point , we use the formula

Using the information from the problem statement, we get 

This simplifies to 

 

Example Question #43 : 3 Dimensional Space

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

Possible Answers:

Correct answer:

Explanation:

To find the equation of a plane with a point  and a normal vector , we use the formula

Using the information from the problem statement, we get

Simplifying, we then get

Example Question #44 : 3 Dimensional Space

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

Possible Answers:

Correct answer:

Explanation:

To find the equation of a plane with a point  and a normal vector , we use the formula

Using the information from the problem statement, we get

Simplifying, we then get

Example Question #45 : 3 Dimensional Space

Find the equation of the plane containing these points:

Possible Answers:

Correct answer:

Explanation:

To find the equation of a plane, we need the normal vector and a point on the plane. The normal vector is found by taking the cross product of two vectors on the plane.

To find two vectors, simply find the difference between terminal and initial points:

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:

Plugging this into the formula

, where  and  is any of the given points (for example ), we get

which simplified becomes

Example Question #41 : 3 Dimensional Space

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

Possible Answers:

Correct answer:

Explanation:

To find the equation of a plane with a point  and normal vector , we use the following equation:

Plugging in the information from the problem statement, we get

Isolating the variables to one side gets us

Example Question #42 : 3 Dimensional Space

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

Possible Answers:

`

Correct answer:

Explanation:

To find the equation of a plane with a point  and normal vector , we use the following equation:

Plugging in the information from the problem statement, we get

Isolating the variables to one side gets us

Example Question #48 : 3 Dimensional Space

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

Possible Answers:

Correct answer:

Explanation:

To find the equation of the plane containing a point  and a normal vector , we use the formula:

Plugging in the known values and solving, we get

Simplifying, we get

 

Example Question #49 : 3 Dimensional Space

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

Possible Answers:

Correct answer:

Explanation:

To find the equation of the plane containing a point  and a normal vector , we use the formula:

Plugging in the known values and solving, we get

Simplifying, we get

 

Example Question #50 : 3 Dimensional Space

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

Possible Answers:

Correct answer:

Explanation:

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

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