Find the domain of the vector function. (Enter your answer in interval notation.)
r(t) = 
t − 1
t + 1
 i + sin(t) j + ln(25 − t2) k

To find the domain of the vector function \( \mathbf{r}(t) = \left( \frac{t-1}{t+1} \right) \mathbf{i} + \sin(t) \mathbf{j} + \ln(25 - t^2) \mathbf{k} \), we need to identify the values of \( t \) for which all three components of the vector are defined.

1. **First component:** \( \frac{t-1}{t+1} \)
   - This is a rational function, so we need to exclude any values of \( t \) that make the denominator zero.
   - Set \( t+1 = 0 \), which gives \( t = -1 \).
   - Therefore, the first component is defined for all \( t \neq -1 \).

2. **Second component:** \( \sin(t) \)
   - The sine function is defined for all real numbers, so there are no restrictions on \( t \) from this component.

3. **Third component:** \( \ln(25 - t^2) \)
   - The natural logarithm function \( \ln(x) \) is only defined when \( x > 0 \).
   - Therefore, \( 25 - t^2 > 0 \), which simplifies to:
     \[
     25 > t^2 \quad \text{or} \quad -5 < t < 5
     \]
   - This means the third component is defined when \( t \in (-5, 5) \).

### Combining the restrictions:
- From the first component: \( t \neq -1 \)
- From the third component: \( t \in (-5, 5) \)

Thus, the domain of \( \mathbf{r}(t) \) is \( (-5, -1) \cup (-1, 5) \).

### Final answer in interval notation:
\[
(-5, -1) \cup (-1, 5)
\]

Would you like further clarification on any part of this solution? 

Here are 5 related questions:
1. How do you find the domain of a rational function?
2. What are the conditions for the domain of a logarithmic function?
3. Can you explain how to find the domain of a vector function in general?
4. How does the sine function affect the domain of a vector function?
5. What would happen if we had an exponential function in the vector function?

**Tip:** Always check for restrictions from all components when finding the domain of a vector function.

Find the domain of the vector function. (Enter your answer in interval notation.) r(t) = (t − 1)/(t + 1) i + sin(t) j + ln(25 − t^2) k

Explore how to find the domain of the vector function r(t) = (t−1)/(t+1) i + sin(t) j + ln(25−t^2) k. This problem involves rational, trigonometric, and logarithmic functions, combining algebraic and calculus-based concepts. Learn how to calculate the domain using interval notation, suitable for college-level students.

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