Area of rectangle having vertices A, B , C and D with position vector
`(-hati + 1/2 hatj + 4hatk) , (hati + 1/2 hatj + 4hatk),(hati - 1/2 hatj + 4hatk)` and `(-hati - 1/2 hatj + 4hatk)` is

Options

A. 1/2 sq. units
B. 1sq. Units
C. 2sq. Units
D. 4sq. Units

Answer

To find the area of the rectangle with vertices A, B, C, and D given by their position vectors, we will follow these steps:

### Step 1: Identify the Position Vectors
The position vectors of the vertices are:
- \( \vec{A} = -\hat{i} + \frac{1}{2}\hat{j} + 4\hat{k} \)
- \( \vec{B} = \hat{i} + \frac{1}{2}\hat{j} + 4\hat{k} \)
- \( \vec{C} = \hat{i} - \frac{1}{2}\hat{j} + 4\hat{k} \)
- \( \vec{D} = -\hat{i} - \frac{1}{2}\hat{j} + 4\hat{k} \)

### Step 2: Calculate the Vector \( \vec{AB} \)
The vector \( \vec{AB} \) is calculated as:
\[ \vec{AB} = \vec{B} - \vec{A} \]
Substituting the position vectors:
\[ \vec{AB} = \left( \hat{i} + \frac{1}{2}\hat{j} + 4\hat{k} \right) - \left( -\hat{i} + \frac{1}{2}\hat{j} + 4\hat{k} \right) \]
\[ = \hat{i} + \frac{1}{2}\hat{j} + 4\hat{k} + \hat{i} - \frac{1}{2}\hat{j} - 4\hat{k} \]
\[ = 2\hat{i} \]

### Step 3: Calculate the Vector \( \vec{BC} \)
The vector \( \vec{BC} \) is calculated as:
\[ \vec{BC} = \vec{C} - \vec{B} \]
Substituting the position vectors:
\[ \vec{BC} = \left( \hat{i} - \frac{1}{2}\hat{j} + 4\hat{k} \right) - \left( \hat{i} + \frac{1}{2}\hat{j} + 4\hat{k} \right) \]
\[ = \hat{i} - \frac{1}{2}\hat{j} + 4\hat{k} - \hat{i} - \frac{1}{2}\hat{j} - 4\hat{k} \]
\[ = -\hat{j} \]

### Step 4: Calculate the Area of the Rectangle
The area \( A \) of the rectangle can be found using the cross product of vectors \( \vec{AB} \) and \( \vec{BC} \):
\[ A = |\vec{AB} \times \vec{BC}| \]
Substituting the vectors:
\[ \vec{AB} = 2\hat{i}, \quad \vec{BC} = -\hat{j} \]
Calculating the cross product:
\[ \vec{AB} \times \vec{BC} = (2\hat{i}) \times (-\hat{j}) = -2(\hat{i} \times \hat{j}) = -2\hat{k} \]

### Step 5: Find the Magnitude of the Cross Product
The magnitude of the cross product gives the area:
\[ A = |-2\hat{k}| = 2 \]

### Final Answer
The area of the rectangle is \( 2 \) square units.