Calculus 3 : Cylindrical Coordinates

Study concepts, example questions & explanations for Calculus 3

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Example Questions

Example Question #61 : Cylindrical Coordinates

A point in space is located, in cylindrical coordinates, at . What is the position of this point in Cartesian coordinates?

Possible Answers:

Correct answer:

Explanation:

If asked to convert cylindrical coordinates of the form  to Cartesian coordinates of the form , it is necessary to relate  and  to the radius, , and the angle, . The relationships are as follows:

Finding  is much simpler; it does not change between Cartesian and cylindrical coordinates:

However, care should be taken when finding  and ; if using a calculator, it is imperative that the correct units (degrees or radians) are specified for the input!

For our coordinates 

 

Example Question #62 : Cylindrical Coordinates

A point in space is located, in cylindrical coordinates, at . What is the position of this point in Cartesian coordinates?

Possible Answers:

Correct answer:

Explanation:

If asked to convert cylindrical coordinates of the form  to Cartesian coordinates of the form , it is necessary to relate  and  to the radius, , and the angle, . The relationships are as follows:

Finding  is much simpler; it does not change between Cartesian and cylindrical coordinates:

However, care should be taken when finding  and ; if using a calculator, it is imperative that the correct units (degrees or radians) are specified for the input!

For our coordinates 

 

Example Question #63 : Cylindrical Coordinates

A point in space is located, in Cartesian coordinates, at . What is the position of this point in cylindrical coordinates?

Possible Answers:

Correct answer:

Explanation:

When given Cartesian coordinates of the form  to cylindrical coordinates of the form , the first and third terms are the most straightforward.

Care should be taken, however, when calculating . The formula for it is as follows: 

However, it is important to be mindful of the signs of both  and , bearing in mind which quadrant the point lies; this will determine the value of :

Quadrants

It is something to bear in mind when making a calculation using a calculator; negative  values by convention create a negative , while negative  values lead to 

For our coordinates 

 

Example Question #64 : Cylindrical Coordinates

A point in space is located, in Cartesian coordinates, at . What is the position of this point in cylindrical coordinates?

Possible Answers:

Correct answer:

Explanation:

When given Cartesian coordinates of the form  to cylindrical coordinates of the form , the first and third terms are the most straightforward.

Care should be taken, however, when calculating . The formula for it is as follows: 

However, it is important to be mindful of the signs of both  and , bearing in mind which quadrant the point lies; this will determine the value of :

Quadrants

It is something to bear in mind when making a calculation using a calculator; negative  values by convention create a negative , while negative  values lead to 

For our coordinates 

 

Example Question #65 : Cylindrical Coordinates

A point in space is located, in Cartesian coordinates, at . What is the position of this point in cylindrical coordinates?

Possible Answers:

Correct answer:

Explanation:

When given Cartesian coordinates of the form  to cylindrical coordinates of the form , the first and third terms are the most straightforward.

Care should be taken, however, when calculating . The formula for it is as follows: 

However, it is important to be mindful of the signs of both  and , bearing in mind which quadrant the point lies; this will determine the value of :

Quadrants

It is something to bear in mind when making a calculation using a calculator; negative  values by convention create a negative , while negative  values lead to 

For our coordinates 

 

Example Question #66 : Cylindrical Coordinates

A point in space is located, in Cartesian coordinates, at . What is the position of this point in cylindrical coordinates?

Possible Answers:

Correct answer:

Explanation:

When given Cartesian coordinates of the form  to cylindrical coordinates of the form , the first and third terms are the most straightforward.

Care should be taken, however, when calculating . The formula for it is as follows: 

However, it is important to be mindful of the signs of both  and , bearing in mind which quadrant the point lies; this will determine the value of :

Quadrants

It is something to bear in mind when making a calculation using a calculator; negative  values by convention create a negative , while negative  values lead to 

For our coordinates 

 

Example Question #67 : Cylindrical Coordinates

A point in space is located, in Cartesian coordinates, at . What is the position of this point in cylindrical coordinates?

Possible Answers:

Correct answer:

Explanation:

When given Cartesian coordinates of the form  to cylindrical coordinates of the form , the first and third terms are the most straightforward.

Care should be taken, however, when calculating . The formula for it is as follows: 

However, it is important to be mindful of the signs of both  and , bearing in mind which quadrant the point lies; this will determine the value of :

Quadrants

It is something to bear in mind when making a calculation using a calculator; negative  values by convention create a negative , while negative  values lead to 

For our coordinates 

 

Example Question #68 : Cylindrical Coordinates

A point in space is located, in Cartesian coordinates, at . What is the position of this point in cylindrical coordinates?

Possible Answers:

Correct answer:

Explanation:

When given Cartesian coordinates of the form  to cylindrical coordinates of the form , the first and third terms are the most straightforward.

Care should be taken, however, when calculating . The formula for it is as follows: 

However, it is important to be mindful of the signs of both  and , bearing in mind which quadrant the point lies; this will determine the value of :

Quadrants

It is something to bear in mind when making a calculation using a calculator; negative  values by convention create a negative , while negative  values lead to 

For our coordinates 

 

Example Question #69 : Cylindrical Coordinates

A point in space is located, in Cartesian coordinates, at . What is the position of this point in cylindrical coordinates?

Possible Answers:

Correct answer:

Explanation:

When given Cartesian coordinates of the form  to cylindrical coordinates of the form , the first and third terms are the most straightforward.

Care should be taken, however, when calculating . The formula for it is as follows: 

However, it is important to be mindful of the signs of both  and , bearing in mind which quadrant the point lies; this will determine the value of :

Quadrants

It is something to bear in mind when making a calculation using a calculator; negative  values by convention create a negative , while negative  values lead to 

For our coordinates 

 

 

Example Question #70 : Cylindrical Coordinates

A point in space is located, in Cartesian coordinates, at . What is the position of this point in cylindrical coordinates?

Possible Answers:

Correct answer:

Explanation:

When given Cartesian coordinates of the form  to cylindrical coordinates of the form , the first and third terms are the most straightforward.

Care should be taken, however, when calculating . The formula for it is as follows: 

However, it is important to be mindful of the signs of both  and , bearing in mind which quadrant the point lies; this will determine the value of :

Quadrants

It is something to bear in mind when making a calculation using a calculator; negative  values by convention create a negative , while negative  values lead to 

For our coordinates 

 

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