Physics - Vectors Question with Solution | TestHub

Minimum additional force needed $\begin{array}{rl}& F=-{\left(F_{\text{resultant }}\right)}_x\\ & F_{\text{resultant }}=[(4-2)(\cos 30\hat{\mathbf{j}}-\sin 30\hat{\mathbf{i}})\\ =& \left[2\left(\frac{\sqrt{3}}{2}\hat{\mathbf{j}}-\frac{1}{2}\hat{\mathbf{i}}\right)+\left(\frac{1}{2}\hat{\mathbf{i}}+\frac{\sqrt{3}}{2}\hat{\mathbf{j}}\right)\right]\\ =& \left[\left(\sqrt{3}+\frac{\sqrt{3}}{2}\right)\hat{\mathbf{j}}+\sin 60\hat{\mathbf{j}}\right)\\ =& \left[-\frac{1}{2}\hat{\mathbf{i}}+\frac{3\sqrt{3}}{2}\hat{\mathbf{i}}+\frac{\hat{\mathbf{i}}}{2}\right)]\\ =& -\frac{\hat{\mathbf{i}}}{2}+\frac{3\sqrt{3}}{2}\hat{\mathbf{j}}F=-\left(-\frac{\hat{\mathbf{i}}}{2}\right)=\frac{1}{2}\hat{\mathbf{i}}\end{array}$ Hence, $|F|=0.5\mathrm{N}$