The sum of two vectors `vec A and vec B` is at right angles to their difference. This is possible if

Options

A. A=2B
B. A=B
C. A=3B
D. B=2A

Explanation

To solve the problem, we need to analyze the condition where the sum of two vectors \(\vec{A}\) and \(\vec{B}\) is at right angles to their difference. Let's break this down step by step.

### Step 1: Understand the vectors involved
We have two vectors:
- The sum: \(\vec{A} + \vec{B}\)
- The difference: \(\vec{A} - \vec{B}\)

### Step 2: Set up the condition for perpendicularity
The condition states that the sum of the vectors is perpendicular to their difference. Mathematically, this can be expressed as:
\[ (\vec{A} + \vec{B}) \cdot (\vec{A} - \vec{B}) = 0 \]
This means that the dot product of the sum and the difference must equal zero.

### Step 3: Calculate the dot product
Now, we will expand the dot product:
\[ (\vec{A} + \vec{B}) \cdot (\vec{A} - \vec{B}) = \vec{A} \cdot \vec{A} - \vec{A} \cdot \vec{B} + \vec{B} \cdot \vec{A} - \vec{B} \cdot \vec{B} \]
Using the property of dot products, we know that \(\vec{A} \cdot \vec{B} = \vec{B} \cdot \vec{A}\). Therefore, we can simplify the expression:
\[ = \vec{A} \cdot \vec{A} - \vec{B} \cdot \vec{B} \]
This can be rewritten as:
\[ = |\vec{A}|^2 - |\vec{B}|^2 \]

### Step 4: Set the equation to zero
Since we know that the dot product must equal zero, we set up the equation:
\[ |\vec{A}|^2 - |\vec{B}|^2 = 0 \]

### Step 5: Solve for the magnitudes of the vectors
From the equation above, we can conclude that:
\[ |\vec{A}|^2 = |\vec{B}|^2 \]
Taking the square root of both sides gives us:
\[ |\vec{A}| = |\vec{B}| \]
This means that the magnitudes of the two vectors are equal.

### Conclusion
Thus, the condition that the sum of two vectors \(\vec{A}\) and \(\vec{B}\) is at right angles to their difference is satisfied if and only if the magnitudes of the two vectors are equal, i.e., \(\vec{A} = \vec{B}\).

---