Final answer:
The equation represents an ellipse in the xy-plane, with the position of a point on the ellipse over time parameterized by trigonometric functions of t and expressed in component vector form.
Step-by-step explanation:
The given equation (x/2)^2 + (y/4)^2 = 1 defines an ellipse ℓ in the xy-plane. This equation describes the set of all points (x, y) such that the sum of the squares of the distances from the x and y axes, divided by the squared lengths of the ellipse's semi-major and semi-minor axes (which are 2 and 4, respectively), is equal to 1. The parametric equations x(t) = 2 cos(t) and y(t) = 4 sin(t) provide a way to generate the points on this ellipse. For a given value of t, which represents the parameter, you can find a corresponding point on the ellipse by plugging t into these equations. The component form ş(t) = x(t)î + y(t)ê is a way to represent the position vector of a point moving along the path of the ellipse over time.