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 calculate the expression \((\vec{A} + \vec{B}) \cdot (\vec{A} - \vec{B})\) where \(\vec{A} = \hat{i} + \hat{j}\) and \(\vec{B} = \hat{i} - \hat{j}\).

### Step-by-Step Solution:

1. **Calculate \(\vec{A} + \vec{B}\)**:
\[ \vec{A} + \vec{B} = (\hat{i} + \hat{j}) + (\hat{i} - \hat{j}) \]
Combine the components:
\[ = \hat{i} + \hat{j} + \hat{i} - \hat{j} = 2\hat{i} \]

2. **Calculate \(\vec{A} - \vec{B}\)**:
\[ \vec{A} - \vec{B} = (\hat{i} + \hat{j}) - (\hat{i} - \hat{j}) \]
Combine the components:
\[ = \hat{i} + \hat{j} - \hat{i} + \hat{j} = 2\hat{j} \]

3. **Calculate the dot product \((\vec{A} + \vec{B}) \cdot (\vec{A} - \vec{B})\)**:
\[ (\vec{A} + \vec{B}) \cdot (\vec{A} - \vec{B}) = (2\hat{i}) \cdot (2\hat{j}) \]
Using the property of dot products, where \(\hat{i} \cdot \hat{j} = 0\):
\[ = 2 \cdot 2 \cdot (\hat{i} \cdot \hat{j}) = 4 \cdot 0 = 0 \]

Thus, the value of \((\vec{A} + \vec{B}) \cdot (\vec{A} - \vec{B})\) is **0**.