Question

If the points \( (k, 1, 5), (1, 0, 3), (7, -2, m) \) are collinear, then \( (k, m) = \)

• A. \( (-2, -1) \)
• B. \( (2, 1) \)
• C. \( (-2, 1) \)
• D. \( (2, -1) \)

Hint:

For three points to be collinear, the vectors formed by the points must be scalar multiples of each other. Solve component-wise to find the relationship.

Answer 1

The Correct Option is A

For three points to be collinear in 3D space, the vectors formed by the points must be scalar multiples of each other. We calculate the vectors \( \overrightarrow{AB} \) and \( \overrightarrow{AC} \) and solve for \( k \) and \( m \). Step 1: Calculate the vectors.
\[ \overrightarrow{AB} = (1 - k, -1, -2), \quad \overrightarrow{AC} = (7 - k, -3, m - 5) \]
Step 2: Set up the scalar multiple condition.
\[ \overrightarrow{AC} = \lambda \overrightarrow{AB} \] Solving the system, we get \( k = -2 \) and \( m = -1 \). Final Answer: \[ \boxed{(-2, -1)} \]