Answer:
The resultant of given forces is [tex]\vec R = -16.099\,\hat{i} -27.002\,\hat{j}\,\,[N][/tex].
Explanation:
The component method consists in summing each component of known vectors in rectangular form to get the resultant. That is:
[tex]\vec R = \left(\Sigma_{i=1}^{m} x_{i}\right)\,\hat{i}+\left(\Sigma_{i=1}^{m} y_{i}\right)\,\hat{j}[/tex]
Where:
[tex]\vec R[/tex] - Resultant, measured in newtons.
[tex]x_{i}[/tex] - i-th x-Component, measured in newtons.
[tex]y_{i}[/tex] - i-th y-Component, measured in newtons.
We describe each known vector below:
[tex]\|\vec F_{1}\| = 12\,N[/tex], South:
[tex]\vec F_{1} = -12\,\hat{i}\,\,[N][/tex]
[tex]\|\vec F_{2}\| = 24\,N[/tex], [tex]\angle = 30^{\circ}[/tex] North of west:
[tex]\vec F_{2} = 24\cdot (-\cos 30^{\circ}\,\hat{i}+\sin 30^{\circ}\,\hat{j})\,[N][/tex]
[tex]\vec F_{2} = -20.785\,\hat{i}+12\,\hat{j}\,\,[N][/tex]
[tex]\|\vec F_{3}\| = 15\,N[/tex], [tex]\angle = 75^{\circ}[/tex] South of west:
[tex]\vec F_{3} = 15\cdot (-\cos 75^{\circ}\,\hat{i}-\sin 75^{\circ}\,\hat{j})\,\,[N][/tex]
[tex]\vec F_{3} = -3.883\,\hat{i}-14.489\,\hat{j}\,\,[N][/tex]
[tex]\|\vec F_{4}\| = 32\,N[/tex], [tex]\angle = 50^{\circ}[/tex] South of east:
[tex]\vec F_{4} = 32\cdot (\cos 50^{\circ}\,\hat{i}-\sin 50^{\circ}\,\hat{j})\,\,[N][/tex]
[tex]\vec F_{4} = 20.569\,\hat{i} -24.513\,\hat{j}\,\,[N][/tex]
We find the resultant by vectorial sum:
[tex]\vec R = \vec F_{1}+\vec F_{2}+\vec F_{3}+\vec F_{4}[/tex]
[tex]\vec R = (-12-20.785-3.883+20.569)\,\hat{i}+(12-14.489-24.513)\,\hat{j}\,\,[N][/tex]
[tex]\vec R = -16.099\,\hat{i} -27.002\,\hat{j}\,\,[N][/tex]
The resultant of given forces is [tex]\vec R = -16.099\,\hat{i} -27.002\,\hat{j}\,\,[N][/tex].
