Asailboat race course consists of four legs, defined by the displacement vectors a, b, c, and d as the drawing indicates. the magnitudes of the first three vectors are a=6.4 km , b=5.10 km, and c=4.80 km. the finish line of the course coincides with the starting line. using the data in the drawing, find the distance of the fourth leg and the angle theta.

The first step is to solve the horizontal component using the given values from the problem and the illustration,
horizontal component = 6.4sin(40°) + 5.1sin(35°) - 4.8sin(23°)

Then the vertical component is determined,
vertical component = 6.4cos(40°) - 5.1cos(35°) - 4.8cos(23°)

The distance of the fourth leg is calculated using the Pythagorean Theorem,
distance = (h²+v²)^1/2 = 6.35 km

Finally, the angle is computed by applying the inverse tangent function,
theta = tan^-1(v/-h) = 54.42°