Question

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To compute the dot product of the vector functions

\vec{r} (t)

and

\vec{u} (t)

, we need to take the dot product of their corresponding components.
The dot product of two vectors

\vec{a} = a_{1} \vec{i} + a_{2} \vec{j} + a_{3} \vec{k}

and

\vec{b} = b_{1} \vec{i} + b_{2} \vec{j} + b_{3} \vec{k}

is given by:

\vec{a} \cdot \vec{b} = a_{1} b_{1} + a_{2} b_{2} + a_{3} b_{3}

Applying this formula to the given vector functions

\vec{r} (t)

and

\vec{u} (t)

:

\vec{r} (t) \cdot \vec{u} (t) = (6 \sin (t)) (10 \sin (t)) + (10 \cos (t)) (6 \cos (t)) + ((t - 15)) (t^{3} - 15)

Simplifying further:

\vec{r} (t) \cdot \vec{u} (t) = 60 \sin^{2} (t) + 60 \cos^{2} (t) + (t - 15) (t^{3} - 15)

Therefore, the dot product of the vector functions

\vec{r} (t)

and

\vec{u} (t)

is given by:

\vec{r} (t) \cdot \vec{u} (t) = 60 \sin^{2} (t) + 60 \cos^{2} (t) + (t - 15) (t^{3} - 15)

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