Question

Determine the resultant of the given vectors by component method. E = 23 km , 11 degrees N of E N = 25 km, 24 degrees E of N G = 19 km, 18 degrees S of W R = 27 km, 58 degrees W of N.

Answer 1

The resultant of the given vectors by component method is
`\vec U = -8.222\,\hat{i}+35.664\,\hat{j}\,\,\,[km]`.

Explanation:

First we define each vector by applying using rectangular form:

1)
`\|\vec E\| = 23\,km`, 11º north of east.

`\vec E = 23\,km\cdot (\cos 11^(\circ)\,\hat{i}+\sin 11^(\circ)\,\hat{j})` (Eq. 1)

2)
`\|\vec N\| = 25\,km`, 24º east of north.

`\vec N = 25\,km\cdot (\sin 24^(\circ)\,\hat{i}+\cos 24^(\circ)\,\hat{j})` (Eq. 2)

3)
`\|\vec G\| = 19\,km`, 18º south of west.

`\vec G = 19\,km\cdot (-\cos 18^(\circ)\,\hat{i}-\sin 18^(\circ)\,\hat{j})` (Eq. 3)

4)
`\|\vec R\| = 27\,km`, 58º west of north.

`\vec R = 27\,km\cdot (-\sin 58^(\circ)\,\hat{i}+\cos 58^(\circ)\,\hat{j})` (Eq. 4)

The resultant of given vectors is determined by vector sum, that is:

`\vec U = \vec E + \vec N + \vec G + \vec R` (Eq. 5)

`\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})`

`\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}`

`\vec U = -8.222\,\hat{i}+35.664\,\hat{j}\,\,\,[km]`

The resultant of the given vectors by component method is
`\vec U = -8.222\,\hat{i}+35.664\,\hat{j}\,\,\,[km]`.