Final answer:
To find points where the position and velocity vectors of a parabola are orthogonal, one must compute the derivative of the position vector, r'(t), and solve r(t) · r'(t) = 0 for t.
Step-by-step explanation:
The question is asking for the points on a parabola at which the position vector r(t) and the velocity vector r'(t) are orthogonal. To find these points, we first need to have a specific representation of the parabola r(t) and then calculate its derivative r'(t). Two vectors are orthogonal if their dot product is zero. Therefore, to find the points where r and r' are orthogonal, we need to solve the equation r(t) · r'(t) = 0. Assuming the function r(t) is given in a Cartesian coordinate system as r(t) = (x(t), y(t)), then r'(t) would be (x'(t), y'(t)). The dot product of r(t) and r'(t) would then be x(t)x'(t) + y(t)y'(t). Setting this equal to zero allows us to find specific values of t at which the position and velocity are orthogonal.