1. The problem: Given 5 vectors, find their resultant vector such that it can land in each of the four quadrants. 2. We first remember that the resultant vector ${\vec R}$ is the vector sum of all given vectors: $$\vec R = \vec v_1 + \vec v_2 + \vec v_3 + \vec v_4 + \vec v_5$$ 3. To ensure that ${\vec R}$ can land in all four quadrants, the sum of vectors must have components $(R_x, R_y)$ that can be positive or negative independently. 4. Let's choose vectors with suitable x and y components: - Vector 1: ${\vec v_1} = (2,3)$ (Quadrant I direction) - Vector 2: ${\vec v_2} = (-4,1)$ (Quadrant II direction) - Vector 3: ${\vec v_3} = (-3,-2)$ (Quadrant III direction) - Vector 4: ${\vec v_4} = (1,-4)$ (Quadrant IV direction) - Vector 5: ${\vec v_5} = (0,0.5)$ (small upward vector) 5. Calculate the resultant vector components: $$R_x = 2 + (-4) + (-3) + 1 + 0 = -4$$ $$R_y = 3 + 1 + (-2) + (-4) + 0.5 = -1.5$$ 6. Current resultant is $(-4,-1.5)$ which lies in Quadrant III (both components negative). 7. To land in other quadrants, try adding or subtracting vectors or adjusting vector directions while keeping 5 total vectors. 8. For Quadrant I ($R_x > 0$, $R_y > 0$), example resultant: $$R_x = 2 + 3 + 1 + 2 + (-1) = 7$$ $$R_y = 3 + 2 + (-1) + 1 + 1 = 6$$ 9. For Quadrant II ($R_x < 0$, $R_y > 0$), example resultant: $$R_x = -5 + (-3) + 1 + 0 + 0 = -7$$ $$R_y = 4 + 2 + (-1) + 0 + 0 = 5$$ 10. For Quadrant IV ($R_x > 0$, $R_y < 0$), example resultant: $$R_x = 3 + 2 + 1 + (-1) + 0 = 5$$ $$R_y = (-3) + (-2) + 0 + 1 + 0 = -4$$ 11. By proper selection and addition of vectors, the resultant vector can indeed be positioned in all four quadrants. Final answer: The resultant vector of 5 vectors can be made to lie in any of the four quadrants by selecting and adding vectors with appropriate components.