To find the domain of the vector-valued function $r(t) = F(t) + G(t)$, we need to analyze the individual functions $F(t)$ and $G(t)$ and identify their domains.

We are given:

$F(t) = \cos(t)i - \sin(t)j + 5t \, k$ $G(t) = 3\cos(t)i + \sin(t)j$

Step 1: Analyze the components of $F(t)$ and $G(t)$

1. $\cos(t)$ and $\sin(t)$: Both $\cos(t)$ and $\sin(t)$ are defined for all real values of $t$. Their domains are $(-\infty, \infty)$.
2. The term $5t \, k$ in $F(t)$: This is just a linear function of $t$, which is also defined for all real $t$.
3. The term $3\cos(t)$ and $\sin(t)$ in $G(t)$: These are just scaled versions of $\cos(t)$ and $\sin(t)$, so they are defined for all real $t$ as well.

Step 2: Determine the domain of $r(t)$

Since both $F(t)$ and $G(t)$ are composed of components that are defined for all real values of $t$, the domain of the vector-valued function $r(t) = F(t) + G(t)$ is the intersection of the domains of $F(t)$ and $G(t)$, which is all real numbers.

Conclusion

The domain of $r(t)$ is $(-\infty, \infty)$.

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Would you like further clarification on any of the steps? Here are some related questions for further study:

1. How do the domains of trigonometric functions affect the domain of vector-valued functions?
2. What happens if one of the components of a vector-valued function has a restricted domain?
3. How can we compute the range of a vector-valued function?
4. What is the difference between the domain of a scalar function and a vector-valued function?
5. How would the domain change if the functions $F(t)$ or $G(t)$ involved square roots or logarithms?

Tip: Always check if any component of the vector-valued function involves a restricted domain (e.g., division by zero, square roots of negative numbers) to avoid overlooking potential domain restrictions.