Question

Find the value of λ, if the points A(−1,−1,2), B(2,8,λ), C(3,11,6) are collinear.

Answer 1

Three points are collinear if the vectors formed by them are parallel.
Let us consider: \[ \vec{AB} = \vec{B} - \vec{A} = (2 - (-1),\ 8 - (-1),\ \lambda - 2) = (3, 9, \lambda - 2) \] \[ \vec{BC} = \vec{C} - \vec{B} = (3 - 2,\ 11 - 8,\ 6 - \lambda) = (1, 3, 6 - \lambda) \]

Since the vectors \( \vec{AB} \) and \( \vec{BC} \) are in the same direction (i.e., collinear), one must be a scalar multiple of the other: \[ \vec{AB} = k \cdot \vec{BC} \] Comparing components: \[ 3 = k \cdot 1 \Rightarrow k = 3 \] \[ 9 = k \cdot 3 = 3 \cdot 3 \Rightarrow \text{(consistent)} \] \[ \lambda - 2 = k \cdot (6 - \lambda) = 3(6 - \lambda) \]

Now solve the equation: \[ \lambda - 2 = 18 - 3\lambda \Rightarrow \lambda + 3\lambda = 18 + 2 \Rightarrow 4\lambda = 20 \Rightarrow \lambda = 5 \]

Final Answer:

\[ \boxed{\lambda = 5} \]