Final answer:
To calculate the magnitude of the car's resultant vector after moving southeast and then in a direction north of east, we use vector addition to sum the east and north/south components, and then apply the Pythagorean theorem.
Step-by-step explanation:
To find the magnitude of the car's resultant vector after traveling 13 km southeast and then 16 km in a direction 40° north of east, we use vector addition. The first vector, representing the southeast direction, can be decomposed into east and south components. The second vector, 16 km at 40° north of east, also has east and north components.
We start by finding the components of each vector using trigonometry:
- The 13 km southeast vector is equivalent to traveling 13 km at a 45° angle south of east, so its east and south components are both 13 cos(45°) km.
- The 16 km vector at 40° north of east has an east component of 16 cos(40°) km and a north component of 16 sin(40°) km.
Then, add the respective east and north/south components. The south component here will be taken as negative since it is in the opposite direction of north. The resultant east component (Reast) is the sum of the east components, and the resultant north component (Rnorth) is the sum of the north component minus the south component (since it is essentially moving north).
Finally, use the Pythagorean theorem to find the magnitude of the resultant vector R, which is given by:
R = √(Reast² + Rnorth²)
The geographic direction can be found by calculating the angle of R relative to the east using the arctangent function.