(a) To find the magnitude of the displacement vector now required to bring the team to the research station, we first have to determine their initial and final positions. The initial position has a x-coordinate of 3.73 km * cos(40.9°), and a y-coordinate of 3.73 km * sin(40.9°). After traveling 1.80 km, their final position has a x-coordinate of 1.80 km * cos(13.8°) and a y-coordinate of 1.80 km * sin(13.8°).
The displacement vector then has an x-component of the x-coordinate of the initial position minus the x-coordinate of the final position, and a y-component of the y-coordinate of the initial position minus the y-coordinate of the final position.
The magnitude of the displacement vector can be found, using the Pythagorean theorem, as the square root of (x-component squared + y-component squared). This gives a magnitude of approximately 2.28 km.
(b) To find the direction of the displacement vector, we first divide the y-component of the displacement vector by the x-component of the displacement vector. Then, we take the arctangent of that quotient to find the direction. However, because the displacement vector can be in any of the four quadrants of a coordinate system, we have to interpret the arctangent's result appropriately. If the x-component is negative, we add 180° to the arctangent's result. If the x-component is positive and the y-component is negative, we add 360° to the arctangent's result.
After all these calculations, we find that the direction of the displacement vector is approximately 61.98° north of east.