Answer to Question #290388 in Physics for AMY

Question

The displacement vectors A and B, when added together, give the resultant vector R, so that R = A + B. Use the data in the drawing and the fact that φ = 31° to find the magnitude R of the resultant vector and the angle θ that it makes with the +x axis.

HOW CAN I SOLVE THIS?

Answer 1

To find the magnitude R of the resultant vector, we need three things: the magnitude $|\vec A|$ of A, the magnitude $|\vec B|$ of B, and the angle $\phi$ between them:

$$A_x=|\vec A|,\\ B_x=|\vec B|\cos\phi,\\ A_y=0,\\ B_y=|\vec B|\sin\phi. \\\space\\ R_x=A_x+B_x=|\vec A|+|\vec B|\cos\phi,\\ R_y=A_y+B_y=|\vec B|\sin\phi,\\\space\\ |\vec R|=\sqrt{(R_x)^2+(R_y)^2}.$$

To find the angle $\theta$ of R above the x-axis:

$$\theta=\arctan\dfrac{R_y}{R_x}=\arctan\dfrac{|\vec B|\sin\phi,}{|\vec A|+|\vec B|\cos\phi}.$$