Question

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Explanation:

Identify the given vectors

We are given two 2D vectors:

$$\vec{U} = 3\hat{i} + \hat{j}$$

$$\vec{V} = 2\hat{i} + 3\hat{j}$$

These represent the adjacent sides of a parallelogram.

Formulate the area equation

The area of a parallelogram spanned by two vectors in a plane is given by the absolute value of the determinant of the matrix formed by the components of the vectors:

$$\text{Area} = |u_x v_y - u_y v_x|$$

Calculate the determinant

Substitute the components \(u_x = 3\), \(u_y = 1\), \(v_x = 2\), and \(v_y = 3\):

$$\text{Area} = |(3)(3) - (1)(2)|$$

$$\text{Area} = |9 - 2| = 7$$

Match with the options

The calculated area is \(7\) square units, which corresponds to option A.

Answer:

• (A) 7 square units (Correct answer)
• (B) 5 square units
• (C) 6 square units
• (D) 3 square units