Unit Tangent Vectors to a Space Curve

Example 1 Find the general formula for the tangent vector and unit tangent vector to the curve given by \(\vec r\left( t \right) = {t^2}\,\vec i + 2\sin t\,\vec j + 2\cos t\,\vec k\).

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Plugging these into the formula for the unit tangent vector, we get???T(t)=\frac{r'(t)}{\left|r'(t)\right|}?????T(t)=\frac{\bold i+2t\bold j}{\sqrt{1+4t^2}}??? Now

ctor!To find the unit tangent vector for a vector function, we use the formula T(t)=(r'(t))/(||r'(t)||), where r'(t) is the derivative of the vector function and t is given. We’ll start

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Find the unit tangent vector for the function r(t)= 2sin(t),4cos(t),4sin2(t) when t= 6π. (A) 173 2,− 2,−2 3 (B) 73 −1,2 3,−3 (c) 113 3,2,−2 2 (D) 191 3,−2,2 3 (E) 171 5,−2 3,−1 Previous question