Area of a rectangle having vertices :
`A (-hat(i)+(1)/(2)hat(j)+4hat(k)), " " B(hat(i)+(1)/(2)hat(j)+4hat(k))`,
`C(hat(i)-(1)/(2)hat(j)+4hat(k)), " " D(-hat(i)-(1)/(2)hat(j)+4hat(k))` is :

Options

A. `(1)/(2)` square unit
B. 1 square unit
C. 2 square units
D. 4 square units

Answer

To find the area of the rectangle defined by the vertices A, B, C, and D, we will follow these steps:

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

### Step 2: Calculate the vectors AB and AD
To find the area of the rectangle, we first need the vectors \( \overrightarrow{AB} \) and \( \overrightarrow{AD} \).

**Vector \( \overrightarrow{AB} \)**:
\[ \overrightarrow{AB} = \overrightarrow{B} - \overrightarrow{A} = \left( \hat{i} + \frac{1}{2}\hat{j} + 4\hat{k} \right) - \left( -\hat{i} + \frac{1}{2}\hat{j} + 4\hat{k} \right) \]
\[ = \hat{i} + \frac{1}{2}\hat{j} + 4\hat{k} + \hat{i} - \frac{1}{2}\hat{j} - 4\hat{k} \]
\[ = (1 + 1)\hat{i} + \left(\frac{1}{2} - \frac{1}{2}\right)\hat{j} + (4 - 4)\hat{k} = 2\hat{i} + 0\hat{j} + 0\hat{k} \]
Thus, \( \overrightarrow{AB} = 2\hat{i} \).

**Vector \( \overrightarrow{AD} \)**:
\[ \overrightarrow{AD} = \overrightarrow{D} - \overrightarrow{A} = \left( -\hat{i} - \frac{1}{2}\hat{j} + 4\hat{k} \right) - \left( -\hat{i} + \frac{1}{2}\hat{j} + 4\hat{k} \right) \]
\[ = -\hat{i} - \frac{1}{2}\hat{j} + 4\hat{k} + \hat{i} - \frac{1}{2}\hat{j} - 4\hat{k} \]
\[ = (-1 + 1)\hat{i} + \left(-\frac{1}{2} - \frac{1}{2}\right)\hat{j} + (4 - 4)\hat{k} = 0\hat{i} - 1\hat{j} + 0\hat{k} \]
Thus, \( \overrightarrow{AD} = -\hat{j} \).

### Step 3: Calculate the cross product \( \overrightarrow{AB} \times \overrightarrow{AD} \)
Now we can find the area of the rectangle using the cross product of \( \overrightarrow{AB} \) and \( \overrightarrow{AD} \):
\[ \overrightarrow{AB} \times \overrightarrow{AD} = (2\hat{i}) \times (-\hat{j}) \]
Using the right-hand rule for the cross product:
\[ = 2(-\hat{k}) = -2\hat{k} \]

### Step 4: Calculate the magnitude of the cross product
The area of the rectangle is given by the magnitude of the cross product:
\[ \text{Area} = |\overrightarrow{AB} \times \overrightarrow{AD}| = |-2\hat{k}| = 2 \]

### Final Answer
Thus, the area of the rectangle is \( 2 \).

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