9514 1404 393
Answer:
Explanation:
The sum of the initial vector and the final vector needs to be equal to the original vector. If we let B represent the vector taken after discovering the mistake, we must have ...
wrong vector + B = intended vector
B = (intended vector) -(wrong vector)
Since the speed is presumed constant, we can write the vectors in terms of hours, not miles. For calculation purposes it is convenient to use bearings measured clockwise from north. West of north will be negative bearing angles for the purpose here. Of course, south is a bearing of 180°, so angles west of south are farther clockwise, hence added to 180°.
B = 4.5∠-22° -0.5∠202° = 4.87207∠-17.91192° . . . using a suitable calculator
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a) The new bearing must be N17.91°W
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b) At the time the error was discovered, the 4.5 hour trip was already 0.5 hours old. Hence the expected additional time is 4.5 -0.5 = 4.0 hours. The actual additional time is 4.872 hours, so the trip will take ...
4.872 -4.0 ≈ 0.9 . . . hours
longer than originally planned.
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Comment on coordinates
Bearings are conventionally rendered as a positive angle in the range 0–360°, measured clockwise from north. Alternatively, they are given (as here) as a positive angle in the range 0–90° east or west from north or south.
Occasionally, you will see bearings expressed as an angle in the range 0–45° from the nearest cardinal direction toward the next-nearest cardinal direction. (For example, E19°S instead of 109° or S71°E.)
Compared to angles measured conventionally on a Cartesian plane, CCW from +x, bearing angles are essentially conventional angles subtracted from 90° (and vice versa). As far as angles are concerned, The Cartesian coordinates are the same as map coordinates if you reflect the map across the line y=x.
Calculators that work with vectors work just fine with bearing distances and angles. You just have to remember that the (a, b) rectangular coordinate pair is (N, E) rather than the usual Cartesian plane (E, N). (East being to the right; North being up.)