The curvature given by the vector function r is k(t)=\frac{|r'(t)\timesr

chillywilly12a

chillywilly12a

Answered question

2021-09-16

The curvature given by the vector function r is
k(t)=|r(t)×r(t)||r(t)|3
Use the formula to find the curvature of r(t)=15t, et, et at the point (0, 1, 1)

Answer & Explanation

dessinemoie

dessinemoie

Skilled2021-09-17Added 90 answers

Step 1
Formula for curvature is k(t)=||r(t)×r(t)||(||r(t)||)3
Find first and second derivatives.
First derivative is r(t)=(15, et, 1et)
Second derivative is r(t)=(0, et, et)
Now, find norm (length) of r(t):||r(t)||=(15)2+(et)2+(1et)2=2cosh(2t)+15
Thus, (||r(t)||)3=(2cosh(2t)+15)32
Next, find cross product of first and second derivatives: /r(t)×r(t)=(2, 15et, 15et
Now, find norm (lenght) of r(t)×r(t):
||r(t)×r(t)||=(2)2+(15et)2+(15et)2=30cosh(2t)+4
Finally, curvature is k(t)=30cosh(2t)+4(2cosh(2t)+15)3=215cosh(2t)+2(2cos

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