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Plane Curves: An Offset Parabola: Example 1
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If you want to see all of the following steps at once, click the "All Steps" button. Otherwise, use the "Next" button.
a) Sketch manually the curve
b) Find
c) Sketch the position vector
and the tangent vector
at
.
How do we proceed to sketch the curve manually?
We choose values of
t
and determine the components..
What are convenient values to use for
t
?
Multiples of 0.5, starting at –2.0.
Fill in the following table. Then check by clicking "Next".
t
x
y
–2.0
–1.5
–1.0
–0.5
0.0
0.5
1.0
1.5
2.0
t
x
y
–2.0
–2.0
2.0
–1.5
–1.5
0.25
–1.0
–1.0
–1.0
–0.5
–0.5
–1.75
0.0
0.0
–2.0
0.5
0.5
–1.75
1.0
1.0
–1.0
1.5
1.5
0.25
2.0
2.0
2.0
Start a plot on paper. Plot the point for
t
= –2.0 Then check by clicking "Next".
Note that the x-component is in blue and the y-component is in green.
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Now add the point for
t
= –1.5
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Plot the remaining points from the table. Then check by clicking "Next".
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Connect the points to show the rough curve. Then check by clicking "Next".
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Here is a smooth version of the full parabola, plotted with many points.
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Now we can find the derivative of
How do we do that?
We take the derivative of each component.
Set that up.
Do the differentiation.
We get
Let's plot the position vector
r
at
t
= 0.25.
Should this be shown as a vector?
Yes.
Where should its tail be placed?
At the origin.
Add
to the last graph above. Then click "Next" to check your work.
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Now consider
. Is this a vector?
Yes.
Where should its tail be?
At the point (head) of
.
Notice that we are considering two properties of a point on the curve:
Its position vector,
, and
the tangent vector,
, at that point.
How do we get the direction of
?
From its components:
Convert to decimals.
Draw the components based at the point.
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Combine the components to get
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What is the name of this vector?
Tangent Vector.
Will it always be along a line tangential to the curve?
Yes.
Will it be perpendicular to the position vector?
No! This is rarely the case.
For example, for a centered circle, the Tangential Vector is perpendicular to the Position Vector, as shown in this diagram:
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Details
The end. If you found this helpful and would recommend that I create more pages like this one, please let me know:
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General Contents
Detailed Contents
Index