Find the unit tangent and unit normal vectors T(t) and N(t). r(t)=&lt;t,3cos⁡t,3sin⁡t&gt;

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Find the unit tangent and unit normal vectors T(t) and N(t). r(t)=&amp;lt;t,3cos⁡t,3sin⁡t&amp;gt;

Mayme · Accepted Answer

Let&#039;s begin with the curve&#039;s given equation, which isr(t)=&amp;lt;t,3cos⁡t,3sin⁡t&amp;gt;Now, calculate the first derivative of the above equation of the curve component-wise, we get,|r′(t)|=12+(−3sin⁡t)2+(3cos⁡t)2=10Calculate the unit tangent vector. It simplifies the later calculations if we leave the vector in this form, with the&amp;nbsp;1x&amp;nbsp;coefficient on the outside of the vector, rather than distributing it within each component of the vector.τ(t)=r′(t)|r′(t)|=110⋅&amp;lt;1,−3sin⁡t,3cos⁡t&amp;gt;Next, let us calculate the first derivative of this unit tangent vector.τ′(t)=110&amp;lt;0,−3cos⁡t,−3sin⁡t&amp;gt;Move the common coefficient of 3 outside the vector to simplify.τ′(t)=310&amp;lt;0,−cos⁡t,−sin⁡t&amp;gt;Calculate the magnitude of&amp;nbsp;τ&amp;nbsp;and simplify|τ′(t)|=(310)2⋅((−cos⁡t)2+(−sin⁡t)2)=310sin2t+cos2t=310(1)=310Simplify after calculating the unit normal vector.N(t)=τ′(t)|τ′(t)|=310&amp;lt;0,−cos⁡t,−sin⁡t&amp;gt;310=&amp;lt;0,−cos⁡t,−sin⁡t&amp;gt;

Find the unit tangent and unit normal vectors T(t) and N(t). r(t)=<t,3\cos t

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