Question

The area of rectangle with sides as \(\vec{A} = 3\hat{i} + 4\hat{j}\) and \(\vec{B} = \hat{i} + 3\hat{j}\) is

• A. \(5\sqrt{10}\) units
• B. 10 units
• C. \(2\sqrt{10}\) units
• D. \(10\sqrt{5}\) units

Hint:

When a question about a geometric shape provides vectors for its sides, it usually implies that the magnitudes of those vectors should be used as the lengths. If the shape was a parallelogram, the area would be given by the magnitude of the cross product, \(|\vec{A} \times \vec{B}|\).

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
We are asked to find the area of a rectangle. The vectors \(\vec{A}\) and \(\vec{B}\) are given to represent the sides. The area of a rectangle is the product of the lengths of its adjacent sides. The wording implies that the lengths of the sides of the rectangle are given by the magnitudes of the vectors \(\vec{A}\) and \(\vec{B}\).

Step 2: Key Formula or Approach:
Area of a rectangle = length \(\times\) width.
The length of a vector \(\vec{V} = x\hat{i} + y\hat{j}\) is its magnitude, given by \(|\vec{V}| = \sqrt{x^2 + y^2}\).

Step 3: Detailed Explanation:
First, we find the length of the side represented by vector \(\vec{A}\).
\[ |\vec{A}| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]
Next, we find the length of the side represented by vector \(\vec{B}\).
\[ |\vec{B}| = \sqrt{1^2 + 3^2} = \sqrt{1 + 9} = \sqrt{10} \]
Now, we calculate the area of the rectangle:
\[ \text{Area} = |\vec{A}| \times |\vec{B}| = 5 \times \sqrt{10} = 5\sqrt{10} \text{ units} \]
Note: For the vectors to represent the sides of a rectangle, they should be perpendicular (\(\vec{A} \cdot \vec{B} = 0\)). Let's check: \(\vec{A} \cdot \vec{B} = (3)(1) + (4)(3) = 15 \neq 0\). The vectors are not perpendicular. The question is poorly phrased, but the intended meaning is to use the magnitudes of the given vectors as the lengths of the rectangle's sides.

Step 4: Final Answer:
The area of the rectangle is \(5\sqrt{10}\) units.