If `veca=hati+hatj,vecb=hati-hatj`, then `veca.vecb=`

Options

A. `(1)/(3)`
B. `(2)/(3)`
C. `(4)/(3)`
D. 0

Answer

To solve the problem, we need to find the dot product of the two vectors \( \vec{a} \) and \( \vec{b} \).

Given:
\[ \vec{a} = \hat{i} + \hat{j} \]
\[ \vec{b} = \hat{i} - \hat{j} \]

### Step 1: Write down the formula for the dot product
The dot product of two vectors \( \vec{a} \) and \( \vec{b} \) is given by:
\[ \vec{a} \cdot \vec{b} = (a_x \cdot b_x) + (a_y \cdot b_y) \]
where \( a_x \) and \( a_y \) are the coefficients of \( \hat{i} \) and \( \hat{j} \) in \( \vec{a} \), and \( b_x \) and \( b_y \) are the coefficients of \( \hat{i} \) and \( \hat{j} \) in \( \vec{b} \).

### Step 2: Identify the coefficients
From \( \vec{a} \) and \( \vec{b} \):
- For \( \vec{a} \):
- \( a_x = 1 \) (coefficient of \( \hat{i} \))
- \( a_y = 1 \) (coefficient of \( \hat{j} \))

- For \( \vec{b} \):
- \( b_x = 1 \) (coefficient of \( \hat{i} \))
- \( b_y = -1 \) (coefficient of \( \hat{j} \))

### Step 3: Substitute the coefficients into the dot product formula
Now we substitute the values into the dot product formula:
\[ \vec{a} \cdot \vec{b} = (1 \cdot 1) + (1 \cdot -1) \]

### Step 4: Calculate the dot product
Calculating the above expression:
\[ \vec{a} \cdot \vec{b} = 1 - 1 = 0 \]

### Conclusion
Thus, the dot product \( \vec{a} \cdot \vec{b} \) is:
\[ \vec{a} \cdot \vec{b} = 0 \]

### Final Answer
The required answer is \( 0 \).

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