Calculus 3 : Normal Vectors

Study concepts, example questions & explanations for Calculus 3

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Example Questions

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Example Question #1 : Normal Vectors

Find the Unit Normal Vector to the given plane. 

.

 

Possible Answers:

Correct answer:

Explanation:

Recall the definition of the Unit Normal Vector.

Let 

 

 

Example Question #2 : Normal Vectors

Find the unit normal vector of .

Possible Answers:

Does not exist

Correct answer:

Does not exist

Explanation:

To find the unit normal vector, you must first find the unit tangent vector.  The equation for the unit tangent vector, ,  is

where  is the vector and  is the magnitude of the vector.

The equation for the unit normal vector,,  is 

where  is the derivative of the unit tangent vector and  is the magnitude of the derivative of the unit vector.

 

For this problem

There is no unit normal vector of .

Example Question #3 : Normal Vectors

Find the unit normal vector of .

Possible Answers:

Does not exist

Correct answer:

Explanation:

To find the unit normal vector, you must first find the unit tangent vector.  The equation for the unit tangent vector, ,  is

where  is the vector and  is the magnitude of the vector.

The equation for the unit normal vector,,  is 

where  is the derivative of the unit tangent vector and  is the magnitude of the derivative of the unit vector.

 

For this problem

Example Question #4 : Normal Vectors

Find the unit normal vector of .

Possible Answers:

DNE

Correct answer:

DNE

Explanation:

To find the unit normal vector, you must first find the unit tangent vector.  The equation for the unit tangent vector, ,  is

where  is the vector and  is the magnitude of the vector.

The equation for the unit normal vector,,  is 

where  is the derivative of the unit tangent vector and  is the magnitude of the derivative of the unit vector.

 

For this problem

The normal vector of  does not exist.

Example Question #5 : Normal Vectors

Find the unit normal vector of .

Possible Answers:

Correct answer:

Explanation:

To find the unit normal vector, you must first find the unit tangent vector.  The equation for the unit tangent vector, ,  is

where  is the vector and  is the magnitude of the vector.

The equation for the unit normal vector,,  is 

where  is the derivative of the unit tangent vector and  is the magnitude of the derivative of the unit vector.

 

For this problem

Example Question #6 : Normal Vectors

Find a normal vector  that is perpendicular to the plane given below.

 

Possible Answers:

No such vector exists.

Correct answer:

Explanation:

Derived from properties of plane equations, one can simply pick off the coefficients of the cartesian coordinate variable to give a normal vector  that is perpendicular to that plane.  For a given plane, we can write

.

From this result, we find that for our case, 

.

Example Question #7 : Normal Vectors

Which of the following is FALSE concerning a vector normal to a plane (in -dimensional space)?

Possible Answers:

It is orthogonal to the plane.

All the other answers are true.

Multiplying it by a scalar gives another normal vector to the plane.

The cross product of any two normal vectors to the plane is .

It is parallel to any other normal vector to the plane.

Correct answer:

All the other answers are true.

Explanation:

These are all true facts about normal vectors to a plane. (If the surface is not a plane, then a few of these no longer hold.)

Example Question #8 : Normal Vectors

Determine whether the two vectors,  and , are orthogonal or not.

Possible Answers:

The two vectors are not orthogonal.

The two vectors are orthogonal.

Correct answer:

The two vectors are orthogonal.

Explanation:

Vectors can be said to be orthogonal, that is to say perpendicular or normal, if their dot product amounts to zero:

To find the dot product of two vectors given the notation

Simply multiply terms across rows:

For our vectors,  and 

The two vectors are orthogonal.

Example Question #9 : Normal Vectors

Determine whether the two vectors,  and , are orthogonal or not.

Possible Answers:

The two vectors are not orthogonal.

The two vectors are orthogonal.

Correct answer:

The two vectors are not orthogonal.

Explanation:

Vectors can be said to be orthogonal, that is to say perpendicular or normal, if their dot product amounts to zero:

To find the dot product of two vectors given the notation

Simply multiply terms across rows:

For our vectors,  and 

The two vectors are not orthogonal.

Example Question #10 : Normal Vectors

Determine whether the two vectors,  and , are orthogonal or not.

Possible Answers:

The two vectors are orthogonal.

The two vectors are not orthogonal.

Correct answer:

The two vectors are not orthogonal.

Explanation:

Vectors can be said to be orthogonal, that is to say perpendicular or normal, if their dot product amounts to zero:

To find the dot product of two vectors given the notation

Simply multiply terms across rows:

For our vectors,  and 

The two vectors are not orthogonal.

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