The magnitude of vectros vec A, vec B` and vec C` are 12, 5 and 13 units respectively and ` vec A + vec B= vec C`, find the angle between ` vec A and vec B` .

Explanation

To find the angle between vectors \( \vec{A} \) and \( \vec{B} \) given that \( \vec{A} + \vec{B} = \vec{C} \) and the magnitudes of these vectors are \( |\vec{A}| = 12 \), \( |\vec{B}| = 5 \), and \( |\vec{C}| = 13 \), we can use the following steps:

### Step 1: Understand the relationship between the vectors
We know that \( \vec{A} + \vec{B} = \vec{C} \). This implies that the vectors \( \vec{A} \) and \( \vec{B} \) can be represented as two sides of a triangle, with \( \vec{C} \) being the resultant vector.

### Step 2: Apply the Law of Cosines
According to the Law of Cosines, for any triangle with sides \( a \), \( b \), and \( c \) opposite to angles \( A \), \( B \), and \( C \) respectively, the following holds:
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