Step 1: Understanding the Concept:
This problem relates vector addition to the magnitudes of the vectors. The first equation is a vector sum, while the second equation relates the squares of the magnitudes in a way that resembles the Pythagorean theorem. We can connect these using the formula for the magnitude of a resultant vector. Step 2: Key Formula or Approach:
The magnitude of the resultant vector $\vec{C} = \vec{A} + \vec{B}$ is given by the law of cosines for vectors:
\[ C^2 = A^2 + B^2 + 2AB \cos\theta \]
where $C = |\vec{C}|$, $A = |\vec{A}|$, $B = |\vec{B}|$, and $\theta$ is the angle between vectors $\vec{A}$ and $\vec{B}$. Step 3: Detailed Explanation:
We are given two conditions:
1. $\vec{A} + \vec{B} = \vec{C}$
2. $A^2 + B^2 = C^2$
From the first condition, we can write the relationship for the magnitudes:
\[ C^2 = A^2 + B^2 + 2AB \cos\theta \]
Now, substitute the second condition into this equation:
\[ A^2 + B^2 = A^2 + B^2 + 2AB \cos\theta \]
Subtract $A^2 + B^2$ from both sides:
\[ 0 = 2AB \cos\theta \]
For this equation to be true, assuming the vectors $\vec{A}$ and $\vec{B}$ are non-zero vectors (i.e., their magnitudes A and B are not zero), the only possibility is:
\[ \cos\theta = 0 \]
The angle $\theta$ for which $\cos\theta = 0$ is $90^\circ$ (or $\frac{\pi}{2}$ radians). Step 4: Final Answer:
The angle between vectors A and B is $90^\circ$. This means the vectors are orthogonal (perpendicular). Therefore, option (C) is correct.
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