## Related questions with answers

Question

Find r(t) · u(t). Is the result a vector-valued function? Explain. r(t) = (3t - 1)i + ¼t³j + 4k, u(t) = t²i - 8j + t³k

Solutions

VerifiedSolution A

Solution B

Answered 9 months ago

Step 1

1 of 5$r(t) = (3t - 1)i + 1/4 * t^{3}j + 4k$, $u(t) = t^{2}i - 8j + t^{3}k$

Original problem

Answered 9 months ago

Step 1

1 of 3To find the value of $\mathbf{r}(t)\cdot\mathbf{u}(t)$, we need to apply the dot product definition which states that if two vectors are given as $\mathbf{r}(t)=x_r(t)\mathbf{i}+y_r(t)\mathbf{j}+z_r(t)\mathbf{k}$ and $\mathbf{u}(t)=x_u(t)\mathbf{i}+y_u(t)\mathbf{j}+z_u(t)\mathbf{k}$, then their dot product is given by:

$\begin{align} \mathbf{r}(t)\cdot\mathbf{u}(t)=x_r(t)x_u(t)+y_r(t)y_u(t)+z_r(t)z_u(t). \end{align}$

Observe that the resultant of the dot product of vectors is a scalar quantity.

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