Two vector-valued functions will parameterize the same tangent line if they yield tangent vectors with the same direction and pass through the same point on the curve. For the same parameter value, their velocity vectors, resulting from the derivative of the position functions, must be parallel or anti-parallel, and their scalar components must correspond.
If two vector-valued functions are found to parameterize the same tangent line, it means that at the point of tangency, both functions yield the same tangent vector which dictates the direction of the tangent line, and they pass through the same point on the curve.
The condition for two vector functions to parameterize the same tangent line is that, for the same value of the parameter (usually time t), the direction ratios (proportional to the coefficients of the unit vectors) of the velocity vectors (derivative of the position functions) must be the same. Essentially, the velocity vectors must be parallel or anti-parallel, which indicates the same slope of the straight line.
The correspondence in scalar components of the two velocity vectors ensures that they define the same line. Whether the vector functions result in exactly the same line depends on whether they also have the same point of application. If they do, the parameterization will describe the same tangent line at that point on the curve. Additionally, if the acceleration vector (second derivative of the position functions) is constant and the vector functions are linear, the vectors will indeed fulfill the criteria for the same tangent line.