A drone is programmed to move according to specific vector instructions. First, it moves according to vector u=(4,3) representing 4 units east and 3 units north. Then, it is instructed to adjust its path using vector v=(-2,1). The final position of the drone can be determined using these vectors. Which of the following statements correctly describes the drone's movement or final position? (Select all that apply.) (C:2)

Question

BlackTom AI · Accepted Answer

The problem gives two movements for the drone: first using vector u = (4, 3), then adjusting with vector v = (-2, 1). To understand the path, I will analyze each answer option in turn before identifying which statements are correct.

Option a: The drone moved 2 units west and 4 units north from its starting position.
- This would correspond to the displacement (-2, 4). However, the actual first move is (4, 3) and the second move is (-2, 1). The cumulative displacement after both moves is u + v = (4, 3) + (-2, 1) = (2, 4). Therefore, option a is not consistent with the given vectors and is incorrect.

Option b: The total distance the drone moved is the magnitude of vector u, which is \sqrt{25}.
- The magnitude of u is |u| = sqrt(4^2 + 3^2) = sqrt(16 + 9) = sqrt(25) = 5. However, the drone moves first along u and then along v, so the total distance traveled is |u| + |v|, not just |u|. Since |v| = sqrt((-2)^2 + 1^2) = sqrt(4 + 1) = sqrt(5). Thus the total distance is 5 + sqrt(5), not simply sqrt(25). This makes option b incorrect.

Option c: The scalar multiplication of 2 with vector v results in a new vector (−4,2), representing 4 units west and 2 units north.
- Scaling v by 2 gives 2*v = 2*(-2, 1) = (-4, 2). This indeed represents 4 units west (negative x) and 2 units north (positive y). Therefore, option c is correct.

Option d: The drone's final position relative to the starting point is represented by the vector (2,4).
- The final position after both movements is the sum of the displacement vectors: u + v = (4, 3) + (-2, 1) = (2, 4). This exactly matches the statement in option d, so option d is correct.

In summary, options c and d align with the vector arithmetic for the given movements, while options a and b misstate the final displacement or total distance traveled.

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