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 ti-\sin tj+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 ti-\cos tj

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

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

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