In the following questions a statement of assertion (A) is followed by a statement of reason ( R).
A : If `vecA bot vecB,` then `|vecA+vecB|=|A-vecB|`.
R: If `vecA bot vecB` then `(vecA+vecB) ` is perpendicular to `vecA-vecB` .

Options

A. If both Assertion & Reason are true and the reason is the correct explanation of the assertion then mark (1) .
B. If both Assertion & Reason are true but the reason is not the correct explanation of the assertion then mark (2)
C. If Assertion is true statement but Reason is false then mark (3).
D. If both Assertion and Reason are false statements then mark (4) .

Answer

To solve the problem, we need to analyze both the assertion (A) and the reason (R) given in the question.

### Step 1: Analyze the Assertion (A)
The assertion states: "If \(\vec{A} \perp \vec{B}\), then \(|\vec{A} + \vec{B}| = |\vec{A} - \vec{B}|\)".

1. **Understanding Perpendicular Vectors**:
If two vectors \(\vec{A}\) and \(\vec{B}\) are perpendicular, the angle \(\theta\) between them is \(90^\circ\).

2. **Using the Magnitude Formula**:
The magnitude of the sum of two vectors is given by:
\[ |\vec{A} + \vec{B}| = \sqrt{|\vec{A}|^2 + |\vec{B}|^2 + 2|\vec{A}||\vec{B}|\cos(\theta)} \]
Since \(\theta = 90^\circ\), \(\cos(90^\circ) = 0\):
\[ |\vec{A} + \vec{B}| = \sqrt{|\vec{A}|^2 + |\vec{B}|^2} \]

3. **Magnitude of the Difference**:
The magnitude of the difference of two vectors is given by:
\[ |\vec{A} - \vec{B}| = \sqrt{|\vec{A}|^2 + |\vec{B}|^2 - 2|\vec{A}||\vec{B}|\cos(\theta)} \]
Again, since \(\theta = 90^\circ\):
\[ |\vec{A} - \vec{B}| = \sqrt{|\vec{A}|^2 + |\vec{B}|^2} \]

4. **Conclusion for Assertion**:
Since both magnitudes are equal:
\[ |\vec{A} + \vec{B}| = |\vec{A} - \vec{B}| \]
Therefore, the assertion (A) is **True**.

### Step 2: Analyze the Reason (R)
The reason states: "If \(\vec{A} \perp \vec{B}\), then \((\vec{A} + \vec{B})\) is perpendicular to \((\vec{A} - \vec{B})\)".

1. **Dot Product for Perpendicularity**:
Two vectors \(\vec{X}\) and \(\vec{Y}\) are perpendicular if their dot product is zero:
\[ \vec{X} \cdot \vec{Y} = 0 \]

2. **Calculate the Dot Product**:
Let \(\vec{X} = \vec{A} + \vec{B}\) and \(\vec{Y} = \vec{A} - \vec{B}\):
\[ \vec{X} \cdot \vec{Y} = (\vec{A} + \vec{B}) \cdot (\vec{A} - \vec{B}) = \vec{A} \cdot \vec{A} - \vec{A} \cdot \vec{B} + \vec{B} \cdot \vec{A} - \vec{B} \cdot \vec{B} \]

3. **Simplifying the Dot Product**:
Since \(\vec{A} \cdot \vec{B} = 0\) (because they are perpendicular):
\[ \vec{X} \cdot \vec{Y} = |\vec{A}|^2 - |\vec{B}|^2 \]
This is not guaranteed to be zero unless \(|\vec{A}| = |\vec{B}|\).

4. **Conclusion for Reason**:
Therefore, the reason (R) is **False** because \((\vec{A} + \vec{B})\) is not necessarily perpendicular to \((\vec{A} - \vec{B})\).

### Final Conclusion
- Assertion (A) is **True**.
- Reason (R) is **False**.

Thus, the correct answer is that the assertion is true but the reason is false.

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