Final answer:
The domain of the vector function r(t) is all real numbers except for t = 0, due to the natural logarithm component which is not defined for t = 0.
Step-by-step explanation:
The domain of the vector function r(t) = 16 - t², e⁻²ᵗ, ln(t²) can be found by considering the domain of each individual component function. For the first component, 16 - t², all real values of t are acceptable since any real number squared is also a real number. For the second component, e⁻²ᵗ, which is an exponential function, all real values of t are also acceptable since exponential functions are defined for all real numbers.
However, for the third component, ln(t²), only values of t such that t² is positive are acceptable, since the natural logarithm function is only defined for positive numbers. Because t² is positive for all t except when t is zero, the domain of ln(t²) is t > 0 or t < 0. Therefore, the domain of the vector function r(t) is all real numbers except for t = 0.