If the position vectors of the points A,B and C be `hati+hatj,hati-hatj` and `ahati+bhatj+chatk` respectively, then the points A,B and C are collinear, if

Options

A. a=b=c=1
B. a=1,b and c are arbitrary scalars
C. ab=c=0
D. c=0,a=1 and b is arbitrary scalars

Explanation

To determine the condition under which the points A, B, and C are collinear given their position vectors, we can follow these steps:

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

### Step 2: Find the vectors AB and BC
To find the vectors AB and BC, we can use the formula for the vector between two points:
- \( \vec{AB} = \vec{B} - \vec{A} \)
- \( \vec{BC} = \vec{C} - \vec{B} \)

Calculating \( \vec{AB} \):
\[ \vec{AB} = (\hat{i} - \hat{j}) - (\hat{i} + \hat{j}) = \hat{i} - \hat{j} - \hat{i} - \hat{j} = -2\hat{j} \]

Calculating \( \vec{BC} \):
\[ \vec{BC} = (a\hat{i} + b\hat{j} + c\hat{k}) - (\hat{i} - \hat{j}) = (a - 1)\hat{i} + (b + 1)\hat{j} + c\hat{k} \]

### Step 3: Set the condition for collinearity
For points A, B, and C to be collinear, the vectors \( \vec{AB} \) and \( \vec{BC} \) must be parallel. This means that there exists a scalar \( k \) such that:
\[ \vec{BC} = k \cdot \vec{AB} \]
Substituting the vectors we found:
\[ (a - 1)\hat{i} + (b + 1)\hat{j} + c\hat{k} = k(-2\hat{j}) \]

### Step 4: Equate components
From the equation above, we can equate the components:
1. For the \( \hat{i} \) component:
\[ a - 1 = 0 \implies a = 1 \]

2. For the \( \hat{j} \) component:
\[ b + 1 = -2k \]

3. For the \( \hat{k} \) component:
\[ c = 0 \]

### Step 5: Conclusion
Thus, the points A, B, and C are collinear if:
- \( a = 1 \)
- \( c = 0 \)
- \( b \) can be any real number (as it depends on the scalar \( k \)).

### Final Condition
The points A, B, and C are collinear if:
- \( a = 1 \)
- \( c = 0 \)
- \( b \) is any real number.