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

Answer

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: Define the vectors
Let:
- \( \mathbf{d_1} = 3\hat{i} + 6\hat{j} - 2\hat{k} \)
- \( \mathbf{d_2} = 4\hat{i} - \hat{j} + 3\hat{k} \)

### Step 2: Calculate the dot product
The dot product \( \mathbf{d_1} \cdot \mathbf{d_2} \) is calculated as follows:
\[ \mathbf{d_1} \cdot \mathbf{d_2} = (3)(4) + (6)(-1) + (-2)(3) \]
Calculating each term:
- \( 3 \times 4 = 12 \)
- \( 6 \times -1 = -6 \)
- \( -2 \times 3 = -6 \)

Now, sum these results:
\[ \mathbf{d_1} \cdot \mathbf{d_2} = 12 - 6 - 6 = 0 \]

### Step 3: Check the magnitudes of the vectors
Calculate the magnitude of \( \mathbf{d_1} \):
\[ |\mathbf{d_1}| = \sqrt{(3^2) + (6^2) + (-2^2)} = \sqrt{9 + 36 + 4} = \sqrt{49} = 7 \]

Now calculate the magnitude of \( \mathbf{d_2} \):
\[ |\mathbf{d_2}| = \sqrt{(4^2) + (-1^2) + (3^2)} = \sqrt{16 + 1 + 9} = \sqrt{26} \]

### Step 4: Analyze the results
- The dot product \( \mathbf{d_1} \cdot \mathbf{d_2} = 0 \), which means the diagonals are perpendicular.
- The magnitudes are \( |\mathbf{d_1}| = 7 \) and \( |\mathbf{d_2}| = \sqrt{26} \), which are not equal.

### Conclusion
Since the diagonals are perpendicular and their magnitudes are not equal, the quadrilateral formed by these diagonals is a **rhombus**.