Final answer:
The domain of the vector function r(t) = 36 - t², e^(-2t), ln(t^5) is all real numbers for the first component, all real numbers for the second component, and t > 0 for the third component.
Step-by-step explanation:
The domain of a vector function is the set of all possible input values that produce a valid output. In this case, the vector function is defined as r(t) = (36 - t²)i + e^(-2t)j + ln(t^5)k.
The domain of this vector function depends on the restrictions on each component. The first component, 36 - t², can take any real value. The second component, e^(-2t), is defined for all real numbers. The third component, ln(t^5), is defined for t > 0 because the natural logarithm is only defined for positive numbers.
Therefore, the domain of the vector function r(t) is all real numbers for the first component, all real numbers for the second component, and t > 0 for the third component.