The point having position vectors `2hati+3hatj+4hatk,3hati+4hatj+2hatk and 4hati+2hatj+3hatk` are the vertices of

Options

A. A. right angled triangle
B. B. isosceles triangle
C. C. equilateral triangle
D. D. collinear

Explanation

To determine the type of triangle formed by the points with position vectors \( \mathbf{A} = 2\hat{i} + 3\hat{j} + 4\hat{k} \), \( \mathbf{B} = 3\hat{i} + 4\hat{j} + 2\hat{k} \), and \( \mathbf{C} = 4\hat{i} + 2\hat{j} + 3\hat{k} \), we will calculate the vectors representing the sides of the triangle and their magnitudes.

### Step 1: Calculate the vectors representing the sides of the triangle

1. **Vector \( \mathbf{AB} \)**:
\[ \mathbf{AB} = \mathbf{B} - \mathbf{A} = (3\hat{i} + 4\hat{j} + 2\hat{k}) - (2\hat{i} + 3\hat{j} + 4\hat{k}) \]
\[ = (3 - 2)\hat{i} + (4 - 3)\hat{j} + (2 - 4)\hat{k} = 1\hat{i} + 1\hat{j} - 2\hat{k} \]

2. **Vector \( \mathbf{BC} \)**:
\[ \mathbf{BC} = \mathbf{C} - \mathbf{B} = (4\hat{i} + 2\hat{j} + 3\hat{k}) - (3\hat{i} + 4\hat{j} + 2\hat{k}) \]
\[ = (4 - 3)\hat{i} + (2 - 4)\hat{j} + (3 - 2)\hat{k} = 1\hat{i} - 2\hat{j} + 1\hat{k} \]

3. **Vector \( \mathbf{AC} \)**:
\[ \mathbf{AC} = \mathbf{C} - \mathbf{A} = (4\hat{i} + 2\hat{j} + 3\hat{k}) - (2\hat{i} + 3\hat{j} + 4\hat{k}) \]
\[ = (4 - 2)\hat{i} + (2 - 3)\hat{j} + (3 - 4)\hat{k} = 2\hat{i} - 1\hat{j} - 1\hat{k} \]

### Step 2: Calculate the magnitudes of the vectors

1. **Magnitude of \( \mathbf{AB} \)**:
\[ |\mathbf{AB}| = \sqrt{(1)^2 + (1)^2 + (-2)^2} = \sqrt{1 + 1 + 4} = \sqrt{6} \]

2. **Magnitude of \( \mathbf{BC} \)**:
\[ |\mathbf{BC}| = \sqrt{(1)^2 + (-2)^2 + (1)^2} = \sqrt{1 + 4 + 1} = \sqrt{6} \]

3. **Magnitude of \( \mathbf{AC} \)**:
\[ |\mathbf{AC}| = \sqrt{(2)^2 + (-1)^2 + (-1)^2} = \sqrt{4 + 1 + 1} = \sqrt{6} \]

### Step 3: Determine the type of triangle

Since all three sides have equal lengths:
\[ |\mathbf{AB}| = |\mathbf{BC}| = |\mathbf{AC}| = \sqrt{6} \]
This means that the triangle formed by the points \( A \), \( B \), and \( C \) is an **equilateral triangle**.

### Final Answer
The points having position vectors \( 2\hat{i} + 3\hat{j} + 4\hat{k} \), \( 3\hat{i} + 4\hat{j} + 2\hat{k} \), and \( 4\hat{i} + 2\hat{j} + 3\hat{k} \) are the vertices of an **equilateral triangle**.