Question

Show that the points \( A(2, 3, 4) \), \( B(-1, -2, 1) \) and \( C(5, 8, 7) \) are collinear.

Hint:

To show that points are collinear, check if the vectors formed by the points are scalar multiples of each other.

Answer 1

To show that the points are collinear, we need to check if the vectors \( \overrightarrow{AB} \) and \( \overrightarrow{AC} \) are scalar multiples of each other. 1. Step 1: Find the vector \( \overrightarrow{AB} \): \[ \overrightarrow{AB} = B - A = (-1, -2, 1) - (2, 3, 4) = (-3, -5, -3) \] 2. Step 2: Find the vector \( \overrightarrow{AC} \): \[ \overrightarrow{AC} = C - A = (5, 8, 7) - (2, 3, 4) = (3, 5, 3) \] 3. Step 3: Check if \( \overrightarrow{AB} \) and \( \overrightarrow{AC} \) are scalar multiples: To check if the vectors are scalar multiples, we compare the ratios of their corresponding components: \[ \frac{-3}{3} = -1, \frac{-5}{5} = -1, \frac{-3}{3} = -1 \] Since the ratios are equal, we conclude that \( \overrightarrow{AB} = -1 \times \overrightarrow{AC} \). Thus, the points \( A \), \( B \), and \( C \) are collinear. Final Answer: The points \( A(2, 3, 4) \), \( B(-1, -2, 1) \), and \( C(5, 8, 7) \) are collinear.