Final answer:
The domain of the vector function is the set of all real numbers except zero and plus/minus 3.
Step-by-step explanation:
The domain of a vector function represents the set of all valid inputs for which the function is defined. In this case, the vector function is given by r(t) = t - 2t² i sin(t) j ln(9 - t²) k. To find the domain, we need to determine the values of t that make the function well-defined.
Since all the components of the vector function have expressions involving t, the domain is determined by the restrictions on t for which these expressions are defined. In this case, we need to consider any values of t that cause division by zero or result in undefined operations, such as taking the logarithm of a negative number.
Therefore, the domain of the vector function r(t) is the set of all real numbers except those that make the expressions t² and 9 - t² equal to zero, as these values would result in undefined operations. So, the domain can be expressed as: t ∈ ℝ, t ≠ 0, t ≠ ±3.