Understanding the Problem:
To solve the problem, we need to find the angle between two vectors given their magnitude and direction. Let's break down the problem step by step.

Step 1: Understanding Vectors:
Vectors are mathematical quantities that have both magnitude (length) and direction. They can be represented graphically as arrows. In this problem, we have three vectors: A, B, and C.

Step 2: Vector Addition:
According to the given information, vector A + vector B = vector C. This implies that the sum of vectors A and B is equal to vector C.

Step 3: Vector Equality:
The equation A + B = C shows that the two vectors are equal. This means that vector A and vector B have the same magnitude and direction as vector C.

Step 4: Applying the Law of Cosines:
To find the angle between vector A and vector B, we can use the Law of Cosines. The Law of Cosines states that in a triangle with sides a, b, and c, the cosine of one angle (let's call it angle C) can be calculated using the formula:

cos(C) = (a^2 + b^2 - c^2) / (2ab)

In this case, the lengths of sides a, b, and c correspond to the magnitudes of vectors A, B, and C, respectively.

Step 5: Solving for the Angle:
To find the angle between vector A and vector B, we can rearrange the Law of Cosines formula as follows:

cos(C) = (|A|^2 + |B|^2 - |C|^2) / (2|A||B|)

Here, |A| represents the magnitude of vector A, |B| represents the magnitude of vector B, and |C| represents the magnitude of vector C.

Step 6: Calculating the Angle:
To obtain the angle between vector A and vector B, we can take the inverse cosine (arccos) of both sides of the equation:

C = arccos((|A|^2 + |B|^2 - |C|^2) / (2|A||B|))

This will give us the angle between vector A and vector B.

Conclusion:
By using the Law of Cosines, we can find the angle between two vectors A and B when their sum is equal to vector C.

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