Answer
$ \mathbf{r}(t) = 2t\mathbf{i} + 2\sin(3t)\mathbf{j} + 5\cos(3t)\mathbf{k}$
Work Step by Step
Step 1
Given:
\[ x = 2t, \quad y = 2\sin(3t), \quad z = 5\cos(3t) \]
Step 2
Then the vector equation of the curve given by: \[ \mathbf{r}(t) = x(t)\mathbf{i} + y(t)\mathbf{j} + z(t)\mathbf{k} \] is: \[ \mathbf{r}(t) = 2t\mathbf{i} + (2\sin(3t))\mathbf{j} + (5\cos(3t))\mathbf{k} \]
Result \[ \mathbf{r}(t) = 2t\mathbf{i} + 2\sin(3t)\mathbf{j} + 5\cos(3t)\mathbf{k} \]
