The position vectors of the points A, B, C are `2 hati + hatj - hatk , 3 hati - 2 hatj + hatk and hati + 4hatj - 3 hatk ` respectively . These points

Options

A. form an isosceles triangle
B. form a right triangle
C. are collinear
D. form a scalene triangle

Explanation

To determine the relationship between the points A, B, and C given their position vectors, we will follow these steps:

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

### Step 2: Calculate the vectors AB and BC
To find the vectors \( \vec{AB} \) and \( \vec{BC} \), we subtract the position vectors:

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

2. **Calculate \( \vec{BC} \)**:
\[ \vec{BC} = \vec{C} - \vec{B} = (\hat{i} + 4\hat{j} - 3\hat{k}) - (3\hat{i} - 2\hat{j} + \hat{k}) \]
\[ = (1 - 3)\hat{i} + (4 + 2)\hat{j} + (-3 - 1)\hat{k} \]
\[ = -2\hat{i} + 6\hat{j} - 4\hat{k} \]

### Step 3: Check for collinearity
To check if the points A, B, and C are collinear, we need to see if the vectors \( \vec{AB} \) and \( \vec{BC} \) are scalar multiples of each other.

We have:
- \( \vec{AB} = \hat{i} - 3\hat{j} + 2\hat{k} \)
- \( \vec{BC} = -2\hat{i} + 6\hat{j} - 4\hat{k} \)

### Step 4: Find a scalar \( \lambda \) such that \( \vec{BC} = \lambda \vec{AB} \)
Let's express \( \vec{BC} \) in terms of \( \vec{AB} \):
\[ \vec{BC} = -2\hat{i} + 6\hat{j} - 4\hat{k} = -2(\hat{i} - 3\hat{j} + 2\hat{k}) \]
This shows that:
\[ \vec{BC} = -2 \cdot \vec{AB} \]
Thus, \( \vec{AB} \) and \( \vec{BC} \) are indeed scalar multiples of each other.

### Conclusion
Since \( \vec{AB} \) and \( \vec{BC} \) are scalar multiples of each other, the points A, B, and C are collinear.

### Final Answer
The points A, B, and C are **collinear**.

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