Prove that F′′(t) has the same direction as a given F(t)

by RoRi

June 1, 2016

Let $A, B$ be nonzero vectors and

$F(t) = e^{2t} A + e^{-2t} B.$

Prove that $F''(t)$ has the same direction as $F(t)$ .

Proof. To show that $F(t)$ and $F''(t)$ have the same direction we must show that there exists a constant $c$ such that $F(t) = cF''(t)$ . So, we compute,

$\begin{align*} F(t) = e^{2t}A + e^{-2t}B && \implies && F'(t) &= 2e^{2t}A - 2 e^{-2t} B \\ && \implies && F''(t) &= 4 e^{2t} A + 4e^{-2t} B \\ && \implies && F(t) &= \frac{1}{4} F''(t). \qquad \blacksquare \end{align*}$

Stumbling Robot

Prove that F′′(t) has the same direction as a given F(t)

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