Step 1: Understanding the Concept:
The area of a rectangle is the product of the lengths of its adjacent sides. Here, the sides are given as vectors. So, we first need to find the magnitudes (lengths) of these two vectors. Step 2: Key Formula or Approach:
The magnitude (length) of a vector $\vec{V} = x\hat{i} + y\hat{j}$ is given by:
\[ |\vec{V}| = \sqrt{x^2 + y^2} \]
The area of the rectangle is given by:
\[ \text{Area} = |\vec{A}| \times |\vec{B}| \]
Step 3: Detailed Explanation:
The vectors representing the sides are:
\[ \vec{A} = 3\hat{i} + 4\hat{j} \]
\[ \vec{B} = \hat{i} + 3\hat{j} \]
First, calculate the magnitude of vector $\vec{A}$:
\[ |\vec{A}| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]
Next, calculate the magnitude of vector $\vec{B}$:
\[ |\vec{B}| = \sqrt{1^2 + 3^2} = \sqrt{1 + 9} = \sqrt{10} \]
The area of the rectangle is the product of the lengths of its sides:
\[ \text{Area} = |\vec{A}| \times |\vec{B}| = 5 \times \sqrt{10} = 5\sqrt{10} \text{ units} \]
Step 4: Final Answer:
The area of the rectangle is $5\sqrt{10}$ units. Therefore, option (A) is correct.
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