You begin your vacation by driving 700 Km west. You then drive 600 Km south, 300 Km east, and 400 Km north. Where do you end up relative to your starting point

 d = 447.21 km  

θ = 26.6º  the two distances are negative, this angle is in the third quadrant

Explanation:

This is a vector addition exercise, the easiest way to solve it is to add the vectors that have the same axis.

East-West (x-axis)

     dₓ = 300 - 700

     dₓ = - 400 km

the negative sign indicates that it is moving west

North-South (y axis)

      = 400 -600

    d_{y} = - 200 km

the negative sign indicates moving towards the South

now we can use the Pythagorean theorem to find the modulus

     d = √ (dₓ² + d_{y}²)

     d = √ (400² + 200²)

     d = 447.21 km

the direction of movement can be found with trigonometry

     tan θ =  d_{y} / dₓ

     θ = tan⁻¹ (200/400)

     θ = 26.6º

Since the two distances are negative, this angle is in the third quadrant

a.

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