If one side of a squre be represented by the vectors `3hati+4hatj+5hatk`, then the area of the square is

Options

A. 12
B. 13
C. 25
D. 50

Explanation

To find the area of a square when one side is represented by the vector \( \vec{a} = 3\hat{i} + 4\hat{j} + 5\hat{k} \), we can follow these steps:

### Step 1: Identify the vector
Let the vector representing one side of the square be:
\[ \vec{a} = 3\hat{i} + 4\hat{j} + 5\hat{k} \]

### Step 2: Calculate the magnitude of the vector
The magnitude (length) of the vector \( \vec{a} \) can be calculated using the formula:
\[ |\vec{a}| = \sqrt{(3)^2 + (4)^2 + (5)^2} \]
Calculating this gives:
\[ |\vec{a}| = \sqrt{9 + 16 + 25} = \sqrt{50} \]

### Step 3: Simplify the magnitude
We can simplify \( \sqrt{50} \):
\[ \sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2} \]
Thus, the length of one side of the square is:
\[ \text{Length of side} = 5\sqrt{2} \]

### Step 4: Calculate the area of the square
The area \( A \) of a square is given by the formula:
\[ A = (\text{side})^2 \]
Substituting the length of the side:
\[ A = (5\sqrt{2})^2 = 25 \times 2 = 50 \]

### Final Answer
The area of the square is:
\[ \boxed{50} \]

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