Curvature via the Vector Cross Product: Trig
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Use the vector cross product method to determine the curvature
.
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.
Simplify with a trig identity.
Equation 2
How do we get
?
We get it by differentiating
Do that.
Next we need the vector cross product
. .
Set up the required matrix.
Do the multiplication.
Simplify.
We get
Find the length of this vector.
Finally, find the curvature by combining this result with equations 1 and 2.
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