If A, B, C are three points with position vectors `i+j,i-j and p.i+qj+rk` respectively, then the points are collinear if

Options

A. `p = q = r =1`
B. `p = q = r = 0`
C. `p =q, r=0`
D. `p =1, q = 2, r= 0`

Answer

To determine the condition for the points A, B, and C to be 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:
- A = \( \mathbf{i} + \mathbf{j} \)
- B = \( \mathbf{i} - \mathbf{j} \)
- C = \( p\mathbf{i} + q\mathbf{j} + r\mathbf{k} \)

### Step 2: Write down the coordinates
From the position vectors, we can extract the coordinates:
- A = (1, 1, 0)
- B = (1, -1, 0)
- C = (p, q, r)

### Step 3: Use the collinearity condition
For points A, B, and C to be collinear, the vectors AB and AC must be parallel. This can be expressed mathematically as:
\[ \mathbf{AB} \times \mathbf{AC} = \mathbf{0} \]
Where:
- \( \mathbf{AB} = B - A = (1 - 1, -1 - 1, 0 - 0) = (0, -2, 0) \)
- \( \mathbf{AC} = C - A = (p - 1, q - 1, r - 0) = (p - 1, q - 1, r) \)

### Step 4: Calculate the cross product
Now we calculate the cross product \( \mathbf{AB} \times \mathbf{AC} \):
\[ \mathbf{AB} \times \mathbf{AC} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 0 & -2 & 0 \\ p - 1 & q - 1 & r \end{vmatrix} \]

Calculating the determinant:
\[ = \mathbf{i} \begin{vmatrix} -2 & 0 \\ q - 1 & r \end{vmatrix} - \mathbf{j} \begin{vmatrix} 0 & 0 \\ p - 1 & r \end{vmatrix} + \mathbf{k} \begin{vmatrix} 0 & -2 \\ p - 1 & q - 1 \end{vmatrix} \]

Calculating each of these determinants:
1. For \( \mathbf{i} \):
\[ = -2r - 0 = -2r \]
2. For \( \mathbf{j} \):
\[ = 0 - 0 = 0 \]
3. For \( \mathbf{k} \):
\[ = 0 - (-2(p - 1)) = 2(p - 1) \]

Thus, the cross product is:
\[ \mathbf{AB} \times \mathbf{AC} = (-2r, 0, 2(p - 1)) \]

### Step 5: Set the cross product to zero
For the vectors to be parallel, we set each component of the cross product to zero:
1. \( -2r = 0 \) → \( r = 0 \)
2. \( 0 = 0 \) (always true)
3. \( 2(p - 1) = 0 \) → \( p - 1 = 0 \) → \( p = 1 \)

### Conclusion
Thus, the points A, B, and C are collinear if:
- \( p = 1 \)
- \( r = 0 \)
- \( q \) can be any real number.

### Final Answer
The points A, B, and C are collinear if \( p = 1 \), \( r = 0 \), and \( q \) can be any real number.

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