A sailboat race course consists of four legs defined by the displacement vectors

Question

A sailboat race course consists of four legs defined by the displacement vectors A, B, C and D shown above.
The values of the angles are T1 = 400, T2 = 390, and T3 = 250.

The magnitudes of the first three vectors are A = 3.5 km, B = 5.1 km and C = 4.6 km. The finish line of the course coincides with the starting line.

The coordinate system for this problem has positive x to the right, positive y as up and counter-clockwise to be a positive angle.
What is the distance of the fourth leg?
What is the value of T4?
What is the value of Dy?

Explanation / Answer

Given that The displacements of boat A along horizontal is Ax = 3.5*cos40 = 2.68 km,
The displacements of boat A along vertical is Ay = 3.5*sin40 = 2.25 km
-------------------------------------------------------------- The displacements of boat B along horizontal is Bx = 5.1*cos39.0 = 3.96 km,
The displacements of boat B along vertical is By = -5.1*sin39.0= -3.21 km --------------------------------------------------------------
The displacements of boat C along horizontal is Cx = 4.6*cos25.0 = -4.17 km,
The displacements of boat C along vertical is Cy = 4.6*sin25.0= -1.94 km --------------------------------------------------------------- --------------------------------------------------------------
The displacements of boat C along horizontal is Cx = 4.6*cos25.0 = -4.17 km,
The displacements of boat C along vertical is Cy = 4.6*sin25.0= -1.94 km ---------------------------------------------------------------
The displacements of boat D along horizontal is Dx = 0 - (Ax+Bx+Cx) = 2.47 km The displacements of boat D along vertical is The displacements of boat D along vertical is The displacements of boat D along vertical is Dy = 0 - (Ay+By+Cy) = -2.9 km

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The distance of the fourth leg is given by, D = sqrt[Dx^2+Dy^2]                                                                     = 3.81 km --------------------------------------------------------------- --------------------------------------------------------------- --------------------------------------------------------------- = arctan(Dy/Dx)    = -40.4 degree So the angle of fourth leg is alon 40.4 degrees with the positive x axis in the clockwise direction.    = -40.4 degree So the angle of fourth leg is alon 40.4 degrees with the positive x axis in the clockwise direction.

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