Find the condition that the three points whose position vectors, `a=ahati+bhatj+chatk,b=hati+chatj and c=-hati-hatj` are collinear.

Answer

To determine the condition for the collinearity of the three points with given position vectors, we can follow these steps:

### Step 1: Identify the Position Vectors
The position vectors of the points are given as:
- \( \vec{A} = a \hat{i} + b \hat{j} + c \hat{k} \)
- \( \vec{B} = \hat{i} + c \hat{j} \)
- \( \vec{C} = -\hat{i} - \hat{j} \)

### Step 2: Find the Vectors \( \vec{AB} \) and \( \vec{BC} \)
To find the vectors \( \vec{AB} \) and \( \vec{BC} \):
- The vector \( \vec{AB} \) is given by:
\[ \vec{AB} = \vec{B} - \vec{A} = (\hat{i} + c \hat{j}) - (a \hat{i} + b \hat{j} + c \hat{k}) = (1 - a) \hat{i} + (c - b) \hat{j} - c \hat{k} \]

- The vector \( \vec{BC} \) is given by:
\[ \vec{BC} = \vec{C} - \vec{B} = (-\hat{i} - \hat{j}) - (\hat{i} + c \hat{j}) = -2 \hat{i} - (1 + c) \hat{j} \]

### Step 3: Set Up the Collinearity Condition
For the points to be collinear, the vectors \( \vec{AB} \) and \( \vec{BC} \) must be parallel. This means there exists a scalar \( \lambda \) such that:
\[ \vec{AB} = \lambda \vec{BC} \]

### Step 4: Write the Equation
Substituting the expressions for \( \vec{AB} \) and \( \vec{BC} \):
\[ (1 - a) \hat{i} + (c - b) \hat{j} - c \hat{k} = \lambda (-2 \hat{i} - (1 + c) \hat{j}) \]

### Step 5: Compare Components
Now we can compare the coefficients of \( \hat{i} \), \( \hat{j} \), and \( \hat{k} \):

1. For \( \hat{i} \):
\[ 1 - a = -2\lambda \quad \text{(1)} \]

2. For \( \hat{j} \):
\[ c - b = -\lambda(1 + c) \quad \text{(2)} \]

3. For \( \hat{k} \):
\[ -c = 0 \quad \text{(3)} \]

### Step 6: Solve the Equations
From equation (3):
\[ c = 0 \]

Substituting \( c = 0 \) into equation (2):
\[ 0 - b = -\lambda(1 + 0) \implies -b = -\lambda \implies \lambda = b \]

Now substituting \( \lambda = b \) into equation (1):
\[ 1 - a = -2b \implies 2b + 1 = a \]

### Conclusion
The condition for the three points to be collinear is:
\[ a = 2b + 1 \]