Calculus 3 : Normal Vectors

Study concepts, example questions & explanations for Calculus 3

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Example Questions

Example Question #61 : 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 #62 : 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 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 #63 : 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.

Example Question #64 : 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 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 #65 : 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.

Example Question #66 : 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 #67 : 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.

Example Question #68 : Normal Vectors

Which of the following vectors is perpendicular to the plane given by the following equation: 

Possible Answers:

Correct answer:

Explanation:

A normal vector to a plane of the form

is given by the gradient of f.  First, we have to put the equation into a form where it equals zero:

The gradient is given by:

A vector multiplied by a constant is parallel to the original vector, so the above vector multiplied by a constant is perpendicular to the plane.  The correct answer is the above vector multiplied by two.

Example Question #69 : Normal Vectors

Find the tangent vector for 

Possible Answers:

Correct answer:

Explanation:

Example Question #70 : Normal Vectors

Find the normal vector (in standard notation) to the plane:

Possible Answers:

Correct answer:

Explanation:

To determine the normal vector to a plane, we simply report the coefficients of the x, y, and z terms, as the equation of a plane is given by

where  is the normal vector.

So, our normal vector is

We were asked to write this in standard notation, which gives us

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