ABC is an isosceles triangle right-angled at | vedclass

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(B) Let the forces be $\vec{F_1} = 6$ along $\overrightarrow{AB}$,$\vec{F_2} = 5$ along $\overrightarrow{CA}$,and $\vec{F_3} = 2\sqrt{2}$ along $\over

Vedclass · Accepted Answer

$5$

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(B) Let the forces be $\vec{F_1} = 6$ along $\overrightarrow{AB}$,$\vec{F_2} = 5$ along $\overrightarrow{CA}$,and $\vec{F_3} = 2\sqrt{2}$ along $\overrightarrow{BC}$.Taking $A$ as the origin $(0,0)$,$\overrightarrow{AB}$ is along the $x$-axis and $\overrightarrow{AC}$ is along the $y$-axis.Since $	riangle ABC$ is an isosceles right-angled triangle,$\angle B = \angle C = 45^\circ$.Vector $\vec{F_1} = 6\hat{i}$.Vector $\vec{F_2} = 5\hat{j}$.Vector $\vec{F_3}$ acts along $\overrightarrow{BC}$. The direction of $\overrightarrow{BC}$ makes an angle of $180^\circ - 45^\circ = 135^\circ$ with the $x$-axis.So,$\vec{F_3} = 2\sqrt{2}(\cos 135^\circ \hat{i} + \sin 135^\circ \hat{j}) = 2\sqrt{2}(-\frac{1}{\sqrt{2}} \hat{i} + \frac{1}{\sqrt{2}} \hat{j}) = -2\hat{i} + 2\hat{j}$.The resultant force $\vec{R} = \vec{F_1} + \vec{F_2} + \vec{F_3} = (6 - 2)\hat{i} + (5 + 2)\hat{j} = 4\hat{i} + 7\hat{j}$.Wait,re-evaluating the direction of $\overrightarrow{CA}$ and $\overrightarrow{BC}$ from the diagram:$\overrightarrow{AB}$ is along $+x$ direction: $\vec{F_1} = 6\hat{i}$.$\overrightarrow{CA}$ is along $-y$ direction: $\vec{F_2} = 5(-\hat{j}) = -5\hat{j}$.$\overrightarrow{BC}$ direction: $\vec{BC} = \vec{AC} - \vec{AB}$. If $AB=AC=a$,then $\vec{B}=(a,0), \vec{C}=(0,a)$. $\vec{BC} = (-a, a)$. Unit vector is $\frac{1}{\sqrt{2}}(-\hat{i} + \hat{j})$.$\vec{F_3} = 2\sqrt{2} \cdot \frac{1}{\sqrt{2}}(-\hat{i} + \hat{j}) = -2\hat{i} + 2\hat{j}$.Resultant $\vec{R} = (6-2)\hat{i} + (-5+2)\hat{j} = 4\hat{i} - 3\hat{j}$.Magnitude $|R| = \sqrt{4^2 + (-3)^2} = \sqrt{16 + 9} = \sqrt{25} = 5$.

$ABC$ is an isosceles triangle right-angled at $A$. Forces of magnitude $2\sqrt{2}$,$5$,and $6$ act along $\overrightarrow{BC}$,$\overrightarrow{CA}$,and $\overrightarrow{AB}$ respectively. The magnitude of their resultant force is

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