VC1.3

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Since i,j,k are constant vectors, \frac{di}{dt}=\frac{dj}{dt}=\frac{dk}{dt}=0\,

Therefore, \frac{dr}{dt}=\frac{d}{dt} (\sin t) i + \frac{d}{dt} (\cos t) j+ \frac{d}{dt} (t) k = \cos t i - \sin t j +k

\frac{d^2 r}{dt^2}=\frac{d}{dt} [\frac{dr}{dt}]=\frac{d}{dt}(\cos t) i - \frac{d}{dt} (\sin t) j + \frac{d}{dt} (1) k = -\sin t i - \cos t j

Therefore, \left |\frac{dr}{dt}\right\vert = \sqrt { (cos t)^2 + (-sin t)^2 +(1)^2}= \sqrt{2}\,

Similarly, \left |\frac{d^2 r}{dt^2}\right\vert = \sqrt { (-sin t)^2 +(-cos t)^2} = 1\,

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