Final answer:
By applying the Law of Cosines to the triangle formed by the two vectors and their difference, it is determined that the angle between vector A and vector B is 120 degrees.
Step-by-step explanation:
The student is trying to find the angle between two vectors vector A and vector B when given their magnitudes and the magnitudes of their sum and difference. By using the Law of Cosines, we can calculate the angle θ between the two vectors. The Law of Cosines relates the lengths of the sides of a triangle to the cosine of one of its angles. In this case, we have a triangle with sides of lengths 4.0 cm, 5.0 cm, and 3.0 cm (the magnitude of the vector difference). The formula for the Law of Cosines in relation to this scenario is:
c^2 = a^2 + b^2 - 2ab*cos(θ),
where 'c' is the length of the side opposite the angle θ, and 'a' and 'b' are the lengths of the other two sides. Applying this to our vector problem with c = 3.0 cm, a = 4.0 cm, and b = 5.0 cm, we can solve for the cosine of angle θ and subsequently determine the angle by taking the inverse cosine (arccos). After calculating, we find that the angle θ is 120 degrees. Thus, the answer is option (d).