Let ABC be a triangle the position vectors of whose vertices are respectively `hati+2hatj+4hatk, -2hati+2hatj+hatk and 2hati+4hatj-3hatk`. Then the `/_\ABC` is (A) isosceles (B) equilateral (C) righat angled (D) none of these

Options

A. isosceles
B. equilateral
C. right angled
D. none of these

Explanation

To determine the type of triangle ABC based on the given position vectors of its vertices, we will follow these steps:

### Step 1: Identify the position vectors
The position vectors of the vertices A, B, and C are given as:
- A: \( \vec{A} = \hat{i} + 2\hat{j} + 4\hat{k} \)
- B: \( \vec{B} = -2\hat{i} + 2\hat{j} + \hat{k} \)
- C: \( \vec{C} = 2\hat{i} + 4\hat{j} - 3\hat{k} \)

### Step 2: Calculate the vectors representing the sides of the triangle
We will calculate the vectors for sides AB, BC, and AC.

- **Vector AB**:
\[ \vec{AB} = \vec{B} - \vec{A} = (-2\hat{i} + 2\hat{j} + \hat{k}) - (\hat{i} + 2\hat{j} + 4\hat{k}) \]
\[ = (-2 - 1)\hat{i} + (2 - 2)\hat{j} + (1 - 4)\hat{k} = -3\hat{i} + 0\hat{j} - 3\hat{k} = -3\hat{i} - 3\hat{k} \]

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

- **Vector AC**:
\[ \vec{AC} = \vec{C} - \vec{A} = (2\hat{i} + 4\hat{j} - 3\hat{k}) - (\hat{i} + 2\hat{j} + 4\hat{k}) \]
\[ = (2 - 1)\hat{i} + (4 - 2)\hat{j} + (-3 - 4)\hat{k} = 1\hat{i} + 2\hat{j} - 7\hat{k} \]

### Step 3: Calculate the magnitudes of the sides
Now we will calculate the magnitudes of the vectors \( \vec{AB} \), \( \vec{BC} \), and \( \vec{AC} \).

- **Magnitude of AB**:
\[ |\vec{AB}| = \sqrt{(-3)^2 + 0^2 + (-3)^2} = \sqrt{9 + 0 + 9} = \sqrt{18} = 3\sqrt{2} \]

- **Magnitude of BC**:
\[ |\vec{BC}| = \sqrt{(4)^2 + (2)^2 + (-4)^2} = \sqrt{16 + 4 + 16} = \sqrt{36} = 6 \]

- **Magnitude of AC**:
\[ |\vec{AC}| = \sqrt{(1)^2 + (2)^2 + (-7)^2} = \sqrt{1 + 4 + 49} = \sqrt{54} = 3\sqrt{6} \]

### Step 4: Check for the type of triangle
To determine if triangle ABC is a right triangle, we can use the Pythagorean theorem. We need to check if the square of the longest side is equal to the sum of the squares of the other two sides.

1. Calculate \( |\vec{AB}|^2 \), \( |\vec{BC}|^2 \), and \( |\vec{AC}|^2 \):
- \( |\vec{AB}|^2 = (3\sqrt{2})^2 = 18 \)
- \( |\vec{BC}|^2 = 6^2 = 36 \)
- \( |\vec{AC}|^2 = (3\sqrt{6})^2 = 54 \)

2. Check if \( |\vec{AC}|^2 = |\vec{AB}|^2 + |\vec{BC}|^2 \):
\[ 54 = 18 + 36 \]
This holds true.

### Conclusion
Since the Pythagorean theorem holds, triangle ABC is a right-angled triangle.

**Final Answer**: (C) right-angled triangle

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