a. Aster is 56.3 m at 3.16° north-east from her initial position
b. She has to head to 183.16° or 86.84° south of west to return to her initial position
<h3>a. How to calculate how far Aster's final position from her initial position?</h3>
Let Aster's initial position be represented by the vector r = 0i + 0j
Since she then walks walks first 70 m in the direction 37° north of east, let this displacement be represented by the vector u = (70sin37°)i + (70cos37°)j
= (70 0.6018)i + (70 0.7986)j
= 42.13i + 55.9j
Also, she then walks 82 m in the direction 20° south of east. Let this displacement be represented by the vector v = (82sin20°)i - (82cos20°)j
= -(82 0.3420)i + (82 0.9397)j
= 28.05i - 77.05j
Finally, she walks 28 m in the direction 30° west of north. Let this displacement be represented by the vector, w = -(28sin30°)i + (28cos30°)j = -(28 0.5)i + (28 0.8660)j
= -14i + 24.25j m
So, the total displacement is R = r + u + v + w
= 42.13i + 55.9j + 28.05i + (-77.05)j + (-14)i + 24.25j m
= 56.18i + 3.1j
So, how far she walks is the magnitude of R. The magnitude of a vector Z = xi + yj is Z = √(x² + y²)
So, the magnitude of R = √((56.18)² + (3.1)²)
= √(3156.19 + 9.61)
= √3165.8
= 56.3 m
<h3>Her direction from final position to initial position</h3>
The direction of a vector Z = xi + yj is given by Ф = tan⁻¹ (y/x)
So, the direction of R is Ф' = tan⁻¹ (3.1/56.18)
= tan⁻¹ (0.0552)
= 3.16°
So, Aster is 56.3 m at 3.16° north-east from her initial position
<h3>b. What direction would she has to head to return to her initial position?</h3>
To return to her original position, the displacement vector is V = r - R
= 0i + 0j - (56.18i + 3.1j)
= -56.18i - 3.1j
So, the direction of V is Ф" = tan⁻¹ (-3.1/-56.18)
= tan⁻¹ (0.0552)
= 3.16°
Since this is in the third quadrant, we have that the direction she must go to return to her original position is α = 180° + 3.16°
= 183.16°
or 90° - 3.16°
= 86.84° south of west
So, she has to head to 183.16° or 86.84° south of west to return to her initial position
Learn more about direction of a vector here:
brainly.com/question/27854247
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