Step 1: Understanding the Question:
The relationship \(\vec{A} + \vec{B} = \vec{C}\) means that \(\vec{C}\) is the resultant of vectors \(\vec{A}\) and \(\vec{B}\).
The magnitudes of the three vectors form a specific relationship which we can test using the parallelogram law of vectors or Pythagorean theorem.
Step 2: Key Formula or Approach:
The magnitude of the resultant is given by \(C^2 = A^2 + B^2 + 2AB \cos \theta\), where \(\theta\) is the angle between \(\vec{A}\) and \(\vec{B}\).
Test if \(A^2 + B^2 = C^2\).
Step 3: Detailed Explanation:
Given magnitudes: \(A = 12\), \(B = 5\), and \(C = 13\).
Let's calculate the squares of these magnitudes:
\[ A^2 = 12^2 = 144 \]
\[ B^2 = 5^2 = 25 \]
\[ C^2 = 13^2 = 169 \]
Checking if they satisfy the Pythagorean triplet condition:
\[ A^2 + B^2 = 144 + 25 = 169 \]
Since \(A^2 + B^2 = C^2\), this implies:
\[ 169 = 169 + 2(12)(5) \cos \theta \]
\[ 0 = 120 \cos \theta \]
\[ \cos \theta = 0 \]
This corresponds to an angle of \(\theta = 90^\circ\).
Step 4: Final Answer:
The vectors form a right-angled triangle where \(\vec{A}\) and \(\vec{B}\) are the legs and \(\vec{C}\) is the hypotenuse. Thus, the angle between \(\vec{A}\) and \(\vec{B}\) is \(90^\circ\).