Consider the given vector equation. r (t) = 2eti + 3e-tj (a) find r' (t). r' (t) = (b) sketch the plane curve together with the position vector r (t) and the tangent vector r' (t) for the given value of t = 0.

(a) Differentiate each of the components to get r' (t). The rule is

... d/dt (a*e^ (bt)) = a*b*e^ (bt)

The answer you have shown is the correct one.

(b) See the figure. The red curve is the position r (t) for 0 ≤ t ≤ 2. The dashed orange line is the tangent line, whose equation is

... L (t) = r (0) + r' (0) * t = (2 + 2t) i + (3 - 3t) j