We are tasked with finding the domain of the vector function $r(t) = 5e^{-t} \mathbf{i} + e^{-t} \mathbf{j} + \ln(t - 1) \mathbf{k}$.

The domain is determined by identifying any restrictions on the input $t$ that would make any of the terms in the function undefined.

1. Exponential terms: Both $5e^{-t}$ and $e^{-t}$ are defined for all real values of $t$. Exponential functions have no restrictions.

2. Logarithmic term: The term $\ln(t - 1)$ is only defined when its argument $t - 1 > 0$, or equivalently: $t > 1$ This restricts $t$ to values greater than 1.

Domain

The domain of $r(t)$ is the set of $t$-values where all components of the function are defined. Since the logarithmic term restricts $t$ to values greater than 1, the domain of $r(t)$ is: $(1, \infty)$

Final Answer:

The domain of $r(t)$ is $(1, \infty)$.

Would you like further details on the logarithmic properties or any other part of the solution?

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Here are 5 related questions to expand on this:

1. What is the range of the logarithmic function $\ln(t - 1)$?
2. How does the behavior of $r(t)$ change as $t \to \infty$?
3. What happens to the value of $e^{-t}$ as $t$ approaches negative infinity?
4. Can you explain how to differentiate the vector function $r(t)$?
5. How does the domain change if the logarithmic term was $\ln(t + 1)$ instead?

Tip: Always check for restrictions in logarithmic functions by ensuring the argument is positive!