If G⃗=〖5t〗2i+tj−t3k and
F⃗=sinti–costj, what is d/dt(G×F⃗) ?
G^=−2i+tj−3tkF^=sinti+costj\hat{G}=-2i+tj-3tk \\ \hat{F}=sin t i+cos t jG^=−2i+tj−3tkF^=sinti+costj
G^×F^=∣ijk−2t−3tsintcost0∣\hat{G}\times \hat{F}=\begin{vmatrix} i &j &k\\-2&t&-3t\\sint &cost &0\end{vmatrix}G^×F^=∣∣i−2sintjtcostk−3t0∣∣
=i(0+3tcost)−j(0+3tsint)+k(−2cost+tsint)=3tcosti−3tsintj+(tsint−2cost)k=i(0+3tcost)-j(0+3tsint)+k(-2cost +tsint)\\[9pt]=3tcosti-3tsint j +(tsint-2cost)k=i(0+3tcost)−j(0+3tsint)+k(−2cost+tsint)=3tcosti−3tsintj+(tsint−2cost)k
⇒d(G^×F^)dt=3(cost−tsint)i−3(sint+tcost)j+(sint+tcost+2sint)k\Rightarrow \dfrac{d(\hat{G}\times \hat{F})}{dt}=3(cost-tsint)i-3(sint+tcost)j+(sint+tcost+2sint)k⇒dtd(G^×F^)=3(cost−tsint)i−3(sint+tcost)j+(sint+tcost+2sint)k
=3(cost−tsint)i−3(sint+tcost)+(3sint+tcost)k=3(cost-tsint)i-3(sint+tcost)+(3sint+tcost)k=3(cost−tsint)i−3(sint+tcost)+(3sint+tcost)k
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