Final answer:
To find the intersection points of the curve r(t) = t i + (3t - t²) k and the paraboloid z = x² y², we equate t² to 0 due to the absence of y in the curve equation, resulting in the origin (0, 0, 0) as the intersection point.
Step-by-step explanation:
The question asks to find the points at which the curve r(t) = t i + (3t - t²) k intersects the paraboloid z = x² y².
To find the intersection points, we need to equate the z-component of the curve to the equation of the paraboloid.
Since the curve is given in vector form with 'i' and 'k' components, we can interpret 'i' as the x-component and 'k' as the z-component.
Therefore, we can equate t² (since z = 3t - t²) to x² y².
We do not have a direct expression for y, but since it does not appear in the equation of our curve, we can consider the cases where y = 0, which would not contribute to z in the equation of the paraboloid.
Thus, our intersection points are found by setting t² = x² * 0² = 0, which gives us t = 0 for the curve.
Substituting back into the curve equation, we have r(0) = 0 i + 0 k, which simplifies to the origin (0, 0, 0) on the paraboloid.