If P, Q, R are three points with respective position vectors `hati + hatj, hati -hatj and a hati + b hatj + c hatk .` The points P, Q, R are collinear, if

Options

A. `a = b = c = 1`
B. `a = b = c = 0`
C. `a=1, b, c in R`
D. `a=1, c =0, b in R`

Answer

To determine the condition for the points P, Q, and R to be collinear given their position vectors, we can follow these steps:

### Step 1: Define the Position Vectors
The position vectors of the points P, Q, and R are given as:
- \( \vec{P} = \hat{i} + \hat{j} \)
- \( \vec{Q} = \hat{i} - \hat{j} \)
- \( \vec{R} = a \hat{i} + b \hat{j} + c \hat{k} \)

### Step 2: Find the Vectors PQ and QR
To check for collinearity, we need to find the vectors \( \vec{PQ} \) and \( \vec{QR} \).

- The vector \( \vec{PQ} \) is given by:
\[ \vec{PQ} = \vec{Q} - \vec{P} = (\hat{i} - \hat{j}) - (\hat{i} + \hat{j}) = \hat{i} - \hat{j} - \hat{i} - \hat{j} = -2\hat{j} \]

- The vector \( \vec{QR} \) is given by:
\[ \vec{QR} = \vec{R} - \vec{Q} = (a \hat{i} + b \hat{j} + c \hat{k}) - (\hat{i} - \hat{j}) = (a - 1) \hat{i} + (b + 1) \hat{j} + c \hat{k} \]

### Step 3: Use the Cross Product Condition
The points P, Q, and R are collinear if the cross product \( \vec{PQ} \times \vec{QR} = \vec{0} \).

Calculating the cross product:
\[ \vec{PQ} \times \vec{QR} = (-2\hat{j}) \times ((a - 1) \hat{i} + (b + 1) \hat{j} + c \hat{k}) \]

Using the determinant method for the cross product:
\[ \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 0 & -2 & 0 \\ a - 1 & b + 1 & c \end{vmatrix} \]

### Step 4: Calculate the Determinant
Calculating the determinant:
\[ \hat{i} \begin{vmatrix} -2 & 0 \\ b + 1 & c \end{vmatrix} - \hat{j} \begin{vmatrix} 0 & 0 \\ a - 1 & c \end{vmatrix} + \hat{k} \begin{vmatrix} 0 & -2 \\ a - 1 & b + 1 \end{vmatrix} \]

Calculating each of these:
1. For \( \hat{i} \):
\[ -2c - 0 = -2c \]
2. For \( \hat{j} \):
\[ 0 - 0 = 0 \]
3. For \( \hat{k} \):
\[ 0 - (-2(a - 1)) = 2(a - 1) \]

Thus, we have:
\[ \vec{PQ} \times \vec{QR} = -2c \hat{i} + 0 \hat{j} + 2(a - 1) \hat{k} \]

### Step 5: Set the Cross Product to Zero
For the vectors to be collinear, we set:
\[ -2c = 0 \quad \text{and} \quad 2(a - 1) = 0 \]

From these equations, we get:
1. \( c = 0 \)
2. \( a - 1 = 0 \) or \( a = 1 \)

### Conclusion
The points P, Q, and R are collinear if:
\[ a = 1 \quad \text{and} \quad c = 0 \]