Final answer:
The correct vector equation of the line with the given parametric equations is (7,2) + s(2,-3), where s belongs to the set of real numbers (R). This vector equation corresponds to answer choice (a).
Step-by-step explanation:
The student asked for the vector equation of the line with given parametric equations x=2t+7, y=−3t+2, where t belongs to the real numbers (R). To convert the given parametric equations into a vector equation, we need to find a position vector and a direction vector. The position vector is a fixed vector that gives the position of a point on the line and the direction vector indicates the direction of the line.
In the equational form, the position vector is equivalent to the point on the line when t=0. Substituting t=0 in the parametric equations, we find the point (7, 2). This is our position vector. The coefficients of t in the parametric equations are the components of the direction vector, which are (2, −3). Therefore, we can write the vector equation of the line as = (7,2) + s(2,−3), where s is a parameter, analogous to t, representing scalar multiples of the direction vector.
The correct answer that represents the vector equation of the line is (a) = (7,2) + s(2,−3), s∈R.