Final answer:
The domain of the vector function r(t) = 9 - t², e⁽⁻⁵ᵗ⁾, ln(t¹) is all real numbers for the first and second terms, and all positive real numbers for the third term.
Step-by-step explanation:
The domain of a vector function represents the values that the input variable, in this case, 't', can take. To find the domain of the vector function r(t) = 9 - t², e⁽⁻⁵ᵗ⁾, ln(t¹), we need to consider any restrictions on the variables in each term.
The first term, 9 - t², has no restrictions since it is defined for all real numbers.
The second term, e⁽⁻⁵ᵗ⁾, is the exponential function and is defined for all real numbers.
The third term, ln(t¹), is the natural logarithm function which is only defined for positive real numbers. Therefore, the domain for the third term is all positive real numbers.
Putting it all together, the domain of the vector function r(t) is all real numbers for the first and second terms, and all positive real numbers for the third term.