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 given question, we need to analyze both the assertion (A) and the reason (R) provided:

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

This means that if vector A is perpendicular to vector B, the magnitudes of the resultant vectors formed by adding and subtracting A and B are equal.

### Step 2: Analyze the Condition of Perpendicular Vectors
When two vectors are perpendicular, the angle \(\theta\) between them is \(90^\circ\). The cosine of \(90^\circ\) is zero.

### Step 3: Apply the Formula for Magnitudes
Using the formula for the magnitude of the sum and difference of two vectors:
- \(|\vec{A} + \vec{B}|^2 = |\vec{A}|^2 + |\vec{B}|^2 + 2|\vec{A}||\vec{B}|\cos(90^\circ)\)
- \(|\vec{A} - \vec{B}|^2 = |\vec{A}|^2 + |\vec{B}|^2 - 2|\vec{A}||\vec{B}|\cos(90^\circ)\)

Since \(\cos(90^\circ) = 0\), we simplify both equations:
- \(|\vec{A} + \vec{B}|^2 = |\vec{A}|^2 + |\vec{B}|^2\)
- \(|\vec{A} - \vec{B}|^2 = |\vec{A}|^2 + |\vec{B}|^2\)

### Step 4: Conclude the Assertion
From the above simplifications, we see that:
\(|\vec{A} + \vec{B}|^2 = |\vec{A} - \vec{B}|^2\)

Taking the square root of both sides gives us:
\(|\vec{A} + \vec{B}| = |\vec{A} - \vec{B}|\)

Thus, the assertion is **true**.

### Step 5: Understand the Reason
The reason states:
**R**: If \(\vec{A} \perp \vec{B}\), then \((\vec{A} + \vec{B})\) is perpendicular to \((\vec{A} - \vec{B})\).

To check this, we can use the dot product:
\[ (\vec{A} + \vec{B}) \cdot (\vec{A} - \vec{B}) = \vec{A} \cdot \vec{A} - \vec{A} \cdot \vec{B} + \vec{B} \cdot \vec{A} - \vec{B} \cdot \vec{B} \]
Since \(\vec{A} \cdot \vec{B} = 0\) (because they are perpendicular), we have:
\[ |\vec{A}|^2 - |\vec{B}|^2 \]
This does not necessarily equal zero unless \(|\vec{A}| = |\vec{B}|\). Therefore, the reason is **not always true**.

### Final Conclusion
- Assertion (A) is **true**.
- Reason (R) is **false**.

Thus, the answer is **C**: Assertion is true, but reason is false.

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