Question

If \(u(t) = \left\langle {sint,cost,t} \right\rangle \) and \(v(t) = \left\langle {t,cost,sint} \right\rangle \), use Formula 4 of Theorem 5 to find\(\frac{d}{{dt}}(u(t).v(t))\).

Short answer

\(2t\cos t - 2\sin t\cos t + 2\sin t\)

Step-by-step solution

Step 1: Assumptions from given data

We have to prove \(\frac{d}{{dt}}(u(t).v(t)) = u'(t).v(t) + u(t).v'(t)\)

\(\begin{array}{l}u(t) = \left\langle {\sin t,\cos t,t} \right\rangle \\u'(t) = \left\langle {\cos t, - \sin t,1} \right\rangle \\v(t) = \left\langle {t,\cos t,\sin t} \right\rangle \\v'(t) = \left\langle {t, - \sin t,\cos t} \right\rangle \end{array}\)

Step 2: Apply the formula

\(\begin{array}{l}\frac{d}{{dt}}(u(t).v(t)) = u'(t).v(t) + u(t).v'(t)\\ = \left\langle {\cos t, - \sin t,1} \right\rangle .\left\langle {t,\cos t,\sin t} \right\rangle + \left\langle {\sin t,\cos t,1} \right\rangle .\left\langle {1, - \sin t,\cos t} \right\rangle \\ = t\cos t - \sin t\cos t + \sin t + \sin t - \sin t\cos t + t\cos t\\ = 2t\cos t - 2\sin t\cos t + 2\sin t\end{array}\)