Consider the diagonals of a quadrilateral formed by the vectors `3hat(i)+6hat(j)-2hat(k) and 4 hat(i)-hat(j)+3hat(k)`. The quadrilateral must be a

Options

A. Square
B. Rhombus
C. Rectangle
D. None of these

Explanation

To determine the type of quadrilateral formed by the diagonals represented by the vectors \( \mathbf{D_1} = 3\hat{i} + 6\hat{j} - 2\hat{k} \) and \( \mathbf{D_2} = 4\hat{i} - \hat{j} + 3\hat{k} \), we will follow these steps:

### Step 1: Calculate the Dot Product of the Vectors
The dot product of two vectors \( \mathbf{A} = a_1\hat{i} + b_1\hat{j} + c_1\hat{k} \) and \( \mathbf{B} = a_2\hat{i} + b_2\hat{j} + c_2\hat{k} \) is given by:

\[ \mathbf{A} \cdot \mathbf{B} = a_1 a_2 + b_1 b_2 + c_1 c_2 \]

For our vectors:

\[ \mathbf{D_1} \cdot \mathbf{D_2} = (3)(4) + (6)(-1) + (-2)(3) \]

Calculating this:

\[ = 12 - 6 - 6 = 0 \]

### Step 2: Calculate the Magnitude of Each Vector
The magnitude of a vector \( \mathbf{A} = a\hat{i} + b\hat{j} + c\hat{k} \) is given by:

\[ |\mathbf{A}| = \sqrt{a^2 + b^2 + c^2} \]

**Magnitude of \( \mathbf{D_1} \)**:

\[ |\mathbf{D_1}| = \sqrt{3^2 + 6^2 + (-2)^2} = \sqrt{9 + 36 + 4} = \sqrt{49} = 7 \]

**Magnitude of \( \mathbf{D_2} \)**:

\[ |\mathbf{D_2}| = \sqrt{4^2 + (-1)^2 + 3^2} = \sqrt{16 + 1 + 9} = \sqrt{26} \]

### Step 3: Analyze the Results
1. The dot product \( \mathbf{D_1} \cdot \mathbf{D_2} = 0 \) indicates that the diagonals are perpendicular.
2. The magnitudes \( |\mathbf{D_1}| = 7 \) and \( |\mathbf{D_2}| = \sqrt{26} \) are not equal.

### Conclusion
Since the diagonals are perpendicular but not equal in length, the quadrilateral formed by these diagonals is a **rhombus**.