Answer:
4.66 km
Explanation:
You want to know how far a driver is from her starting position after traveling 5.67 km north, 2.21 km southwest, then 3.77 km east.
Resultant
The sum of the vectors is shown in the two attachments.
The first attachment shows the direction measured CCW from east. The angle in the second attachment is the bearing angle from the origin, measured CW from north. In both cases, the distance is 4.66 km.
Vectors
In (north, east) coordinates, the distance traveled on each vector is ...
5.67 km, bearing 0° : (5.67, 0)
2.21 km, bearing 225° : (-1.5627, -1.5627)
3.77 km, bearing 90° : (0, 3.77)
Then the coordinates of the final position are ...
(5.67 -1.5627 +0, 0 -1.5627 +3.77) = (4.1073, 2.2073)
The length of this vector is √(4.1073² +2.2073²) ≈ √21.7421 ≈ 4.66
She is about 4.66 km from where she started.
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Additional comment
We sometimes find it convenient to use bearing angles and (north, east) coordinates in problems involving navigation and compass directions. If you want to use more conventional (east, north) coordinates, and angles measured CCW from the +x axis, you can swap the rectangular coordinates, and subtract the bearing angles from 90°.
The conversions are (x, y) = r(cos(θ), sin(θ)), and r = √(x²+y²).
The two attachments show the different angle measures. The angle 62° north of east in the first attachment is the same as the bearing angle 28° east of north in the second attachment.
