Welcome to Westonci.ca, where your questions are met with accurate answers from a community of experts and enthusiasts. Connect with a community of professionals ready to help you find accurate solutions to your questions quickly and efficiently. Join our platform to connect with experts ready to provide precise answers to your questions in different areas.
Sagot :
The unit tangent vector is T(u) and the unit normal vector is N(t) if the vector function. R(t) is equal to 9 2 t, e9t, e−9t.
What is vector?
It is defined as the quantity that has magnitude as well as direction also the vector always follows the sum triangle law.
We have vectored function:
[tex]\rm R(t) = (9\sqrt{2t}, e^{9t}, e^{-9t})[/tex]
Find its derivative:
[tex]\rm R'(t) = (9\sqrt{2}, 9e^{9t}, -9e^{-9t})[/tex]
Now its magnitude:
[tex]\rm |R'(t) |= \sqrt{(9\sqrt{2})^2+ (9e^{9t})^2+ (-9e^{-9t})^2}[/tex]
After simplifying:
[tex]\rm R'(t) = 9 \dfrac{e^{18t}+1}{e^{9t}}[/tex]
Now the unit tangent is:
[tex]\rm T(u) = \dfrac{R'(t)}{|R'(t)|}[/tex]
After dividing and simplifying, we get:
[tex]\rm T(u) = \dfrac{1}{e^{18t}+1} (\sqrt{2}e^{9t}, e^{18t}, -1)[/tex]
Now, finding the derivative of T(u), we get:
[tex]\rm T'(u) = \dfrac{1}{(e^{18t}+1)^2} (9\sqrt{2}e^{9t}(1-e^{18t}), 18e^{18t}, 18e^{18t})[/tex]
Now finding its magnitude:
[tex]\rm |T'(u) |= \dfrac{1}{(e^{18t}+1)^2} (9\sqrt{2}e^{9t}(1-e^{18t})^2+ (18e^{18t})^2+( 18e^{18t})^2)[/tex]
After simplifying, we get:
[tex]\rm |T'(u)|= \dfrac{9\sqrt{2}e^{9t}}{e^{18t}+1}[/tex]
Now for the normal vector:
Divide T'(u) and |T'(u)|
We get:
[tex]\rm N(t) = \dfrac{1}{e^{18t}+1} ( 1-e^{18t}, \sqrt{2}e^{9t}, \sqrt{2}e^{9t})[/tex]
Thus, the unit tangent vector is T(u) and the unit normal vector is N(t) if the vector function. R(t) is equal to 9 2 t, e9t, e−9t.
Learn more about the vector here:
https://brainly.com/question/8607618
#SPJ4
Thanks for using our platform. We aim to provide accurate and up-to-date answers to all your queries. Come back soon. We hope our answers were useful. Return anytime for more information and answers to any other questions you have. Your questions are important to us at Westonci.ca. Visit again for expert answers and reliable information.
