Functions, Graphs, and Limits
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AP Calculus BC › Functions, Graphs, and Limits
Convert the following cartesian coordinates into polar form:
Explanation
Cartesian coordinates have and
, represented as
. Polar coordinates have
is the hypotenuse, and
is the angle.
Solution:
What is the polar form of ?
Explanation
We can convert from rectangular to polar form by using the following trigonometric identities: and
. Given
, then:
Dividing both sides by , we get:
What is the vector form of ?
Explanation
Given , we need to map the
,
, and
coefficients back to their corresponding
,
, and
-coordinates.
Thus the vector form of is
.
What is the vector form of ?
Explanation
In order to derive the vector form, we must map the ,
,
-coordinates to their corresponding
,
, and
coefficients.
That is, given, the vector form is
.
So for , we can derive the vector form
.
What is the vector form of ?
Explanation
In order to derive the vector form, we must map the ,
,
-coordinates to their corresponding
,
, and
coefficients.
That is, given , the vector form is
.
So for , we can derive the vector form
.
What is the vector form of ?
Explanation
In order to derive the vector form, we must map the ,
,
-coordinates to their corresponding
,
, and
coefficients.
That is, given, the vector form is
.
So for , we can derive the vector form
.
Given points and
, what is the vector form of the distance between the points?
Explanation
In order to derive the vector form of the distance between two points, we must find the difference between the ,
, and
elements of the points. That is, for any point
and
, the distance is the vector
.
Subbing in our original points and
, we get:
Given and
, what is
in terms of
(rectangular form)?
Explanation
Knowing that and
, we can isolate
in both equations as follows:
Since both of these equations equal , we can set them equal to each other:
Given and
, what is
in terms of
(rectangular form)?
None of the answers provided
Explanation
Given and
, let's solve both equations for
:
Since both equations equal , let's set them equal to each other and solve for
:
What is the polar form of ?
Explanation
We can convert from rectangular to polar form by using the following trigonometric identities: and
. Given
, then:
Dividing both sides by , we get: