Question

If a person walks first 70 m in the direction 37° north of east, and then walks 82 m in the

Answer 1

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

a. How to calculate how far Aster's final position from her initial position?

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

Her direction from final position to initial position

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

b. What direction would she has to head to return to her initial position?​

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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