Final answer:
The domain of the vector function r(t) = (t - 1)ti + sin(t)j + ln(25 - t²)k is (-5, 5) in interval notation.
Step-by-step explanation:
To find the domain of the vector function r(t) = (t - 1)ti + sin(t)j + ln(25 - t²)k, we need to determine the values of t for which the function is defined. The function is defined as long as the natural logarithm term ln(25 - t²) is defined. The natural logarithm is only defined for positive arguments, so we set 25 - t² > 0 and solve for t.
25 - t² > 0
t² < 25
-5 < t < 5
The domain of the vector function is therefore (-5, 5) in interval notation.