The area of the square, one of whose diagonals is `3hat(i)+ 4hat(j)` is:

Options

A. 12 square unit
B. 12.5 square unit
C. 25 square unit
D. 156.25 square unit

Answer

To find the area of the square whose diagonal is given as \( \vec{D} = 3\hat{i} + 4\hat{j} \), we can follow these steps:

### Step 1: Find the magnitude of the diagonal vector
The magnitude of the diagonal vector \( \vec{D} \) can be calculated using the formula:

\[ |\vec{D}| = \sqrt{(3)^2 + (4)^2} \]

### Step 2: Calculate the squares
Calculating the squares of the components:

\[ |\vec{D}| = \sqrt{9 + 16} \]

### Step 3: Simplify the expression
Now, simplify the expression inside the square root:

\[ |\vec{D}| = \sqrt{25} \]

### Step 4: Find the length of the diagonal
Taking the square root gives us:

\[ |\vec{D}| = 5 \]

### Step 5: Use the diagonal to find the area of the square
The area \( A \) of a square can be calculated using the formula:

\[ A = \frac{1}{2} \times (\text{diagonal})^2 \]

Substituting the length of the diagonal:

\[ A = \frac{1}{2} \times (5)^2 \]

### Step 6: Calculate the area
Calculating the area:

\[ A = \frac{1}{2} \times 25 = \frac{25}{2} = 12.5 \]

### Final Answer
Thus, the area of the square is \( 12.5 \) square units.

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