What is the area of the rectangle of which `vec(r)=ahat(i)+b hat(j)` is a semidiagonal ?

Options

A. `a^(2)+b^(2)`
B. `2(a^(2)+b^(2))`
C. `4(a^(2)+b^(2))`
D. `4ab`

Explanation

To find the area of the rectangle for which the vector \(\vec{r} = a \hat{i} + b \hat{j}\) is a semi-diagonal, we can follow these steps:

### Step 1: Understand the semi-diagonal
The given vector \(\vec{r} = a \hat{i} + b \hat{j}\) represents a semi-diagonal of the rectangle. A semi-diagonal is half of the full diagonal of the rectangle.

### Step 2: Find the full diagonal vector
Since \(\vec{r}\) is the semi-diagonal, the full diagonal vector \(\vec{D}\) can be expressed as:
\[ \vec{D} = 2\vec{r} = 2(a \hat{i} + b \hat{j}) = 2a \hat{i} + 2b \hat{j} \]

### Step 3: Identify the length and breadth
From the diagonal vector \(\vec{D} = 2a \hat{i} + 2b \hat{j}\), we can identify:
- Length of the rectangle = \(2a\)
- Breadth of the rectangle = \(2b\)

### Step 4: Calculate the area of the rectangle
The area \(A\) of a rectangle is given by the formula:
\[ A = \text{Length} \times \text{Breadth} \]
Substituting the values we found:
\[ A = (2a) \times (2b) = 4ab \]

### Final Answer
Thus, the area of the rectangle is:
\[ \boxed{4ab} \]