If the angle between two vectors `vecA and vecB " is " 90^(@)` then

Options

A. `vecA = 2vecB `
B. `vecA - vecB = vec0`
C. `|vecA + vecB| = |vecA -vecB| `
D. `vecA + vecB = vec0`

Answer

To solve the problem, we need to analyze the implications of having two vectors, \(\vec{A}\) and \(\vec{B}\), that are at a right angle (90 degrees) to each other.

### Step-by-Step Solution:

1. **Understanding the Condition**:
Given that the angle between vectors \(\vec{A}\) and \(\vec{B}\) is \(90^\circ\), we know that they are orthogonal. This means that the dot product of the two vectors is zero.
\[ \vec{A} \cdot \vec{B} = |\vec{A}| |\vec{B}| \cos(90^\circ) = 0 \]

2. **Analyzing Option A**:
The first option states that \(\vec{A} = 2\vec{B}\). This implies that \(\vec{A}\) and \(\vec{B}\) are not only different in magnitude but also suggests a specific relationship in terms of their lengths. However, since the angle between them is \(90^\circ\), this statement does not hold true as it does not contradict the orthogonality condition. Thus, this option is **incorrect**.

3. **Analyzing Option B**:
The second option states that \(\vec{A} - \vec{B} = 0\), which implies that \(\vec{A} = \vec{B}\). If this were true, the angle between them would be \(0^\circ\) (they would be the same vector), which contradicts the given condition of \(90^\circ\). Therefore, this option is also **incorrect**.

4. **Analyzing Option C**:
The third option states that \(|\vec{A} + \vec{B}| = |\vec{A} - \vec{B}|\). To analyze this, we can use the formula for the magnitude of the sum and difference of two vectors:
\[ |\vec{A} + \vec{B}| = \sqrt{|\vec{A}|^2 + |\vec{B}|^2 + 2|\vec{A}||\vec{B}|\cos(90^\circ)} = \sqrt{|\vec{A}|^2 + |\vec{B}|^2} \]
\[ |\vec{A} - \vec{B}| = \sqrt{|\vec{A}|^2 + |\vec{B}|^2 - 2|\vec{A}||\vec{B}|\cos(90^\circ)} = \sqrt{|\vec{A}|^2 + |\vec{B}|^2} \]
Since both magnitudes are equal, this option is **correct**.

5. **Analyzing Option D**:
The fourth option states that \(\vec{A} + \vec{B} = 0\), which implies that \(\vec{A} = -\vec{B}\). If this were true, the angle between them would be \(180^\circ\), which contradicts the given condition of \(90^\circ\). Thus, this option is **incorrect**.

### Conclusion:
The only correct option is **Option C**: \(|\vec{A} + \vec{B}| = |\vec{A} - \vec{B}|\).