Final answer:
The angle between the displacement vector and the first vector for the hiker who moves 3 meters east and then 4 meters north would be 36.87 degrees. However, given the closest multiple-choice option, the answer would likely be taken as (b) 45 degrees.
Step-by-step explanation:
The student is asking about the angle between the displacement vector and the first vector of a hiker's path. Since the hiker moves 3 meters east (which can be considered along the x-axis) and then 4 meters north (along the y-axis), we can use pythagorean theorem to consider the displacement as the hypotenuse of a right triangle where these two paths are perpendicular to each other.
The displacement vector is, therefore, directly diagonal from the starting point, making a right triangle with the 3 meter and 4 meter sides. The angle in question is the angle between the eastward vector (vector 1) and the displacement vector (vector 3). The eastward vector corresponds to the base of the triangle and the displacement vector is the hypotenuse. Since this is a 3-4-5 right triangle, the angle opposite the 4 meter side with respect to the 3 meter base is 53.13 degrees in trigonometric terms. However, since the question seems to be looking for the smallest angle between the vectors, which is the angle adjacent to the 4 meter side, we subtract this from 90 degrees to find the angle of interest. Hence, 90 - 53.13 degrees equals 36.87 degrees, which isn't an option here. However, assuming a possible typo in the options, the closest correct answer would be 45 degrees if rounded to nearest option.
Thus, the closest answer to the multiple-choice options provided would likely be (b) 45 degrees, despite none of the options exactly matching the calculated angle of 36.87 degrees.