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Question

1. A diver explores a shallow reef off the coast of Belize. She initially swims d1 = 82.9 m north, makes a turn to the east and continues for d2 = 177.8 m, then follows a big grouper (a kind of fish) for d3 = 84.1 m in the direction θ = 41.7° north of east. Finally, she swims d4 = 193.9 m south.

1. A) What is the straight-line distance between her initial and final positions in m?  

1. B) Suppose she wants to swim back to her initial position in a straight line. In what direction should she swim? Express your answer as an angle in degrees north of west.  

2. Consider the two vectors,

A = 16.5x + 35.5y

B = 5.7x - 21y

2. A) What is the magnitude of the vector A - B? 

2. B) What is the angle of the vector A - B, measured counterclockwise in degrees from the positive x-axis? 

2. C) What is the magnitude of the vector 3A + B? 

2. D) What is the angle of the vector 3A+B, measured counterclockwise in degrees from the positive x-axis? 

Answer 1

Taking i and j as unit distances in the directions east and north respectively we have:

(d1) 0i + 82.9j

(d2) 177.8i + 0j

(d3) 84.1cos41.7°i + 84.1sin41.7°j

(d4) 0i - 193.9j

Total i => 240.6 m

Total j => -55.1 m

so, |d| = √(240.6)² + (-55.1)² => 246.8 m

Angle => tan⁻¹(55.1/240.6) = 13°

Hence, direction 13° North of West

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