Aboat travels 800 m east, then 200 m south, then 600 m at a direction of 50° south of west. 1) what is the distance from where the boat started to where it finished?
2) what is the direction from where the boat started to where it finished, measured south of east?

1) The distance is 779 m.

2) The direction is 58° south of east.

Explanation:

1) To find the distance we have to sum the vectors and then find the magnitude of the resulting vector. Please, see the figure for a better understanding of the situation.

The y-component of the vector A is 0 (because it is directed horizontally relative to the origin of the frame of reference). Then, the vector A must be:

A = (800 m, 0)

The vector B has an x-component of 0, then the vector B will be:

B = (0, -200 m)

Notice that we consider the direction east and north as positive.

To find the components of the vector C we have to use trigonometry. Notice that the vector C is the hypotenuse of the right triangle formed by cx, cy, and C. Then, by trigonometry:

cos 50° = cx/C

C · cos 50° = cx

The magnitude of the x-component of the vector C is:

cx = 600 m · cos 50° = 386 m

In the same way for the y-component of C

sin 50° = cy/C

cy = 600 · sin 50° = 460 m

Then, the vector C will be: C = (-386 m, -460 m)

The resulting vector of the sum of each vector will be:

R = A + B + C = (800 m, 0) + (0, -200 m) + (-386 m, -460 m)

R = (800 m + 0 m - 386 m, 0 m -200 m - 460 m)

R = (414 m, -660 m)

The magnitude of the vector R will be:

R² = (414 m)² + (-660 m)²

R = 779 m

The distance from where the boat started to where it finished is 779 m.

2) To find the direction, we have to use trigonometry again. Please see the figure to notice that the x-component of R is 779 m · cos θ and the y-component is 779 m · sin θ  (remember: cos θ = adjacent / hypotenuse and sin θ =  opposite/hypotenuse).

Then:

414 m = 779 m · cos θ

cos θ = 414 m / 779 m

θ = 58°

Then, the direction of the resulting vector is 58° south of east

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