Curvature via the Vector Cross Product: Parabola
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Use the vector cross product method to determine thecurvature
.
How do we proceed?
We use
Equation 1
This involves vectors, the cross product, and derivatives. Which do we do first?
We need the derivatives first.
How do we get
?
We differentiate each component of
Set that up.
Do the differentiation.
We'll also need the length,
. How do we get it?
The length is the square root of the sum of the squares of the components.
Find this length.
Equation 2
How do we get
?
We get it by differentiating
Do that.
Next we need the vector cross product
. With so few non-zero components involved, let’s do the cross product directly, without using the usual matrix.
Set that up.
We'll have to find
.
What is the value of
?
The cross product of any vector with itself is always zero.
What is the value of
?
Use this information to get the cross product we need.
Find the length of this vector.
Finally, find the curvature by combining this result with equations 1 and 2.
Here is a graph of this parabola.
(Notice that the curve is always in the plane z = 2.)
Where does the graph show the most curvature? At t = 0 or at t = 3?
At t = 0.
Evaluate the curvature at t = 0.
Evaluate the curvature at t = 3.
These values confirm our visual inspection.
The end. If you found this helpful and would recommend that I create more pages like this one, please let me know: Email to John Taylor