Final answer:
Point P (1, 2) creates a partition along the directed line segment from A (0, 0) to B (4, 8) in a 1:3 ratio. The lengths of vectors AP and PB indicate that for every unit from A to P, there are three units from P to B.
Step-by-step explanation:
The question is asking to find the ratio in which the point P (1, 2) partitions the directed line segment AB, with A being (0, 0) and B being (4, 8). We can approach this by calculating the vectors AP and PB. The vector AP is the difference between the coordinates of P and A, which gives us AP = P - A = (1 - 0, 2 - 0) = (1, 2). Similarly, PB = B - P = (4 - 1, 8 - 2) = (3, 6). The lengths of these vectors are proportional to the distances between the points. The length of AP (using the vector components) is 1 unit in the x-direction and 2 units in the y-direction, whereas the length of vector PB is 3 times the x-component and 3 times the y-component of AP. Therefore, point P divides the line segment AB in the ratio 1:3. This means that for every 1 unit from A to P, there are 3 units from P to B.