Determine the resultant of the following vector system: 65 km at 60 degrees south of east, 120 km at 30 degrees west of north, 30 km at 35 degrees east of north, and 75 km south.

To determine the resultant of the vector system described, it is essential to break each individual vector down into its components. By analyzing each vector, one can sum the components in the east-west direction and the north-south direction separately to find the overall magnitude and direction of the resultant vector.

The first vector, 65 km at 60 degrees south of east, can be split into components: the east component is (65 \cdot \cos(60^\circ)) and the south component is (65 \cdot \sin(60^\circ)).

The second vector, 120 km at 30 degrees west of north, has an east component that is negative: ( -120 \cdot \sin(30^\circ)) (since it goes west) and a north component of (120 \cdot \cos(30^\circ)).

For the third vector, 30 km at 35 degrees east of north, the east component is (30 \cdot \sin(35^\circ)) and the north component is (30 \cdot \cos(35^\circ)).

Finally, the fourth vector is straightforward: 75 km south has no east component (0) and contributes -75 km