If `vec(A)=hati +hat j` and `vec(B)=hati-hatj`
The value of `(vec(A)+vec(B)).(vec(A)-vec(B))` is :

Options

A. `sqrt(2)`
B. 0
C. `(1)/(2)`
D. 2

Explanation

To solve the problem, we need to compute the expression \((\vec{A} + \vec{B}) \cdot (\vec{A} - \vec{B})\) given the vectors \(\vec{A} = \hat{i} + \hat{j}\) and \(\vec{B} = \hat{i} - \hat{j}\).

### Step 1: Calculate \(\vec{A} + \vec{B}\)
\[ \vec{A} + \vec{B} = (\hat{i} + \hat{j}) + (\hat{i} - \hat{j}) \]
Combine like terms:
\[ = \hat{i} + \hat{j} + \hat{i} - \hat{j} = 2\hat{i} + 0\hat{j} = 2\hat{i} \]

### Step 2: Calculate \(\vec{A} - \vec{B}\)
\[ \vec{A} - \vec{B} = (\hat{i} + \hat{j}) - (\hat{i} - \hat{j}) \]
Combine like terms:
\[ = \hat{i} + \hat{j} - \hat{i} + \hat{j} = 0\hat{i} + 2\hat{j} = 2\hat{j} \]

### Step 3: Compute the dot product \((\vec{A} + \vec{B}) \cdot (\vec{A} - \vec{B})\)
Now we need to calculate:
\[ (2\hat{i}) \cdot (2\hat{j}) \]
Using the property of dot product:
\[ = 2 \cdot 2 \cdot (\hat{i} \cdot \hat{j}) \]
Since \(\hat{i} \cdot \hat{j} = 0\) (because they are perpendicular):
\[ = 4 \cdot 0 = 0 \]

### Final Answer
The value of \((\vec{A} + \vec{B}) \cdot (\vec{A} - \vec{B})\) is **0**.

---