Calculus 3 : Cylindrical Coordinates

Study concepts, example questions & explanations for Calculus 3

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Example Questions

Example Question #21 : 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 

 (Bearing in mind sign convention)

Example Question #22 : 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 

 (Bearing in mind sign convention)

Example Question #23 : 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 

 (Bearing in mind sign convention)

Example Question #24 : 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 

 (Bearing in mind sign convention)

Example Question #25 : 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 

 (Bearing in mind sign convention)

Example Question #26 : 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 

 (Bearing in mind sign convention)

Example Question #21 : 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:

When given cylindrical coordinates of the form  and converting to Cartesian coordinates of the form , the relationship between , and  will be of use:

 (This value is identical across Cartesian and cylindrical coordinate systems)

For the cylindrical coordinates

The Cartesian coordinates are

Example Question #22 : 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:

When given cylindrical coordinates of the form  and converting to Cartesian coordinates of the form , the relationship between , and  will be of use:

 (This value is identical across Cartesian and cylindrical coordinate systems)

For the cylindrical coordinates 

The Cartesian coordinates are

Example Question #29 : 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:

When given cylindrical coordinates of the form  and converting to Cartesian coordinates of the form , the relationship between , and  will be of use:

 (This value is identical across Cartesian and cylindrical coordinate systems)

For the cylindrical coordinates 

The Cartesian coordinates are

Example Question #30 : 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:

When given cylindrical coordinates of the form  and converting to Cartesian coordinates of the form , the relationship between , and  will be of use:

 (This value is identical across Cartesian and cylindrical coordinate systems)

For the cylindrical coordinates 

When finding the Cartesian coordinates, be sure if using a calculator to have the correct angle units set (degrees vs radians):

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