Can you find the resultant force between these two vectors?

Question

Determine the magnitude of the resultant force on an object if force $A$ is pulling the object with $150$ lbs of force and force $B$ is pulling with $300$ lbs, and the angle between the two forces is $110^\circ$.

Answer 1

use this formula of triangle $a^2=b^2+c^2-2bc\cdot \cos A $ where a,b,c are sides of triangle.

so resultant force $$\vec R^2={150}^2+{300}^2-2\cdot150\cdot 300 \cos 70^\circ$$ $$\vec R=\sqrt {22500+90000-30781.81}$$ $$\vec R=285.86\,lbs$$

There was typo in the formula which I have now corrected. a^2=b^2+c^2 - 2bc * cos A

Answer 2

If we assume $f_A$ points in the positive $x$ direction and $f_B$ is located in quadrant II, we have,

$$f_A = \left[ \begin{array}{c} 150\\ 0 \end{array} \right]$$

and

$$f_B = \left[ \begin{array}{c} 300\cos(110) \\ 300\sin(110) \end{array} \right]$$

The resultant is

$$f_R = f_A + f_B = \left[ \begin{array}{c} 150 + 300\cos(110) \\ 300\sin(110) \end{array} \right]$$

then

$$\|f_R\| = \sqrt{(150 + 300\cos(110))^2 + (300\sin(110))^2}=285.86$$

Note: Notice that $\cos(110)$ is negative.