Final answer:
To find the direction of the third leg of the journey, we can calculate the angle between the displacement vectors of the second and final positions. Using the dot product formula, we find that the angle is approximately 12.2 degrees. Therefore, the direction of the third leg is 12.2 degrees south of east.
Step-by-step explanation:
In order to find the direction of the third leg of the journey, we need to consider the displacement vector of each leg. The first leg is 2.00 km east, which can be represented as a vector (2.00, 0) km. The second leg is 3.50 km southeast, which can be represented as a vector (3.50*cos(45), -3.50*sin(45)) km. The final position is 5.80 km directly east of the starting point, so the displacement vector of the third leg is (5.80, 0) km.
To find the direction of the third leg, we can find the angle between the two vectors: (5.80, 0) km and (3.50*cos(45), -3.50*sin(45)) km. Using the dot product formula: cos(theta) = (A dot B) / (|A| * |B|), we can calculate the angle theta. Rearranging the formula, we get theta = acos((A dot B) / (|A| * |B|)). Plugging in the values, we get theta = acos(((5.80)(3.50*cos(45))) / ((5.80)(3.50))). Evaluating this expression, we find that the angle theta is approximately 12.2 degrees. Therefore, the direction of the third leg of the journey is 12.2 degrees south of east.