Final answer:
The domain of the vector function r(t) = 25 - t², e⁽⁻⁴ᵗ⁾, ln(t¹) is t > 0.
Step-by-step explanation:
To find the domain of the vector function r(t) = 25 - t², e⁽⁻⁴ᵗ⁾, ln(t¹), we need to look at the restrictions on each component of the function.
The first component, 25 - t², is a polynomial function and is defined for all real numbers.
The second component, e⁽⁻⁴ᵗ⁾, is an exponential function, which is defined for all real numbers.
The third component, ln(t¹), is a logarithmic function, and it is defined only for positive real numbers. Therefore, the domain of the vector function is t > 0.