If points A(2, \(\beta\), 3 ), B(\(\alpha\), -5, 1) and C(-1, 11, 9) are collinear, then what is the value of \(\alpha + \beta\) ?

2

\(\vec{AB} = \vec{OB} - \vec{OA} \) 

\(\vec{AB} = (\alpha - 2)\hat{i} - (5 + \beta)\hat{j} - 2\hat{k}\)

\(\vec{AC} = \vec{OC} - \vec{OA} \) 

\(\vec{AB} = - 3\hat{i} + (11 - \beta)\hat{j} + 6\hat{k}\)

\(\vec{AB}\) & \(\vec{AC}\) are collinear ⇒ \(\vec{AB} = \lambda \vec{AC}\) ; \(\lambda \neq 0\)

\(\alpha\) - 2 = - 3 \(\lambda\)

- 5 - \(\beta\) = \(\lambda (11 - \beta)\)

- 2 = 6 \(\lambda\)

⇒ \(\lambda = - \frac{1}{3} ; \alpha = 3 ; \beta = -1 \)

\(\alpha + \beta = 3 - 1 = 2\)