Vector A has a magnitude of 3.0 units and makes an angle of −90.0º with the positive x-axis. Vector B has a magnitude of 4.0 units and makes an angle of −120º with the positive x-axis. What is the direction of the vector sum A + B relative to the positive x-axis?

Question

Quizplus · Accepted Answer

To find the direction of the vector sum $A + B$, we first need to break each vector into its x and y components and then sum these components. For vector A:- $A_x = 3.0 \cos(-90^\circ) = 0$ (since $\cos(90^\circ) = 0$)- $A_y = 3.0 \sin(-90^\circ) = -3.0$ (since $\sin(-90^\circ) = -1$)For vector B:- $B_x = 4.0 \cos(-120^\circ) = 4.0 \times \cos(120^\circ) = 4.0 \times (-0.5) = -2.0$- $B_y = 4.0 \sin(-120^\circ) = 4.0 \times \sin(120^\circ) = 4.0 \times (\sqrt{3}/2) = 2\sqrt{3}$Summing the components:- $Sum_x = A_x + B_x = 0 - 2.0 = -2.0$- $Sum_y = A_y + B_y = -3.0 + 2\sqrt{3}$The direction $\theta$ of the resultant vector relative to the positive x-axis is given by $\theta = \tan^{-1}\left(\frac{Sum_y}{Sum_x}\right)$.Plugging in the values:- $\theta = \tan^{-1}\left(\frac{-3.0 + 2\sqrt{3}}{-2.0}\right)$This calculation gives a value around -107 degrees, indicating that the vector sum points in a direction of approximately -107º relative to the positive x-axis.

Vector a Has a Magnitude of 3