Answer:
The resultant of the given vectors by component method is [tex]\vec U = -8.222\,\hat{i}+35.664\,\hat{j}\,\,\,[km][/tex].
Step-by-step explanation:
First we define each vector by applying using rectangular form:
1) [tex]\|\vec E\| = 23\,km[/tex], 11º north of east.
[tex]\vec E = 23\,km\cdot (\cos 11^{\circ}\,\hat{i}+\sin 11^{\circ}\,\hat{j})[/tex] (Eq. 1)
2) [tex]\|\vec N\| = 25\,km[/tex], 24º east of north.
[tex]\vec N = 25\,km\cdot (\sin 24^{\circ}\,\hat{i}+\cos 24^{\circ}\,\hat{j})[/tex] (Eq. 2)
3) [tex]\|\vec G\| = 19\,km[/tex], 18º south of west.
[tex]\vec G = 19\,km\cdot (-\cos 18^{\circ}\,\hat{i}-\sin 18^{\circ}\,\hat{j})[/tex] (Eq. 3)
4) [tex]\|\vec R\| = 27\,km[/tex], 58º west of north.
[tex]\vec R = 27\,km\cdot (-\sin 58^{\circ}\,\hat{i}+\cos 58^{\circ}\,\hat{j})[/tex] (Eq. 4)
The resultant of given vectors is determined by vector sum, that is:
[tex]\vec U = \vec E + \vec N + \vec G + \vec R[/tex] (Eq. 5)
[tex]\vec U = 23\,km\cdot (\cos 11^{\circ}\,\hat{i}+\sin 11^{\circ}\,\hat{j})+25\,km \cdot (\sin 24^{\circ}\,\hat{i}+\cos 24^{\circ}\,\hat{j})+19\,km\cdot (-\cos 18^{\circ}\,\hat{i}-\sin 18^{\circ}\,\hat{j})+27\,km\cdot (-\sin 58^{\circ}\,\hat{i}+\cos 58^{\circ}\,\hat{j})[/tex]
[tex]\vec U = (23\,km\cdot \cos 11^{\circ}+25\,km\cdot \sin 24^{\circ}-19\,km\cdot \cos 18^{\circ}-27\,km\cdot \sin 58^{\circ})\,\hat{i}+(23\,km\cdot \sin 11^{\circ}+25\,km\cdot \cos 24^{\circ}-19\,km\cdot \sin 18^{\circ}+27\,km\cdot \cos 58^{\circ})\,\hat{j}[/tex]
[tex]\vec U = -8.222\,\hat{i}+35.664\,\hat{j}\,\,\,[km][/tex]
The resultant of the given vectors by component method is [tex]\vec U = -8.222\,\hat{i}+35.664\,\hat{j}\,\,\,[km][/tex].
