Using vectors, find the value of 'b' if the points A(-$\displaystyle 1$, -$\displaystyle 1$, $\displaystyle 2$), B($\displaystyle 2$, b, $\displaystyle 5$) and C($\displaystyle 3$, $\displaystyle 11$, $\displaystyle 6$) are collinear . Also, determine the ratio in which the point B divides the line-segment AC internally.

Let \(\displaystyle \vec{a}, \vec{b}, \vec{c}\) be the position vectors of points \(\displaystyle A, B, C\) respectively \[\begin{aligned} & \vec{a}=-\hat{\imath}-\hat{\jmath}+2 \hat{k} \\ & \vec{b}=2 \hat{\imath}+b \hat{\jmath}+5 \hat{k} \\ & \vec{c}=3 \hat{\imath}+11 \hat{\jmath}+6 \hat{k} \\ & \overrightarrow{A B}=\vec{b}-\vec{a}=3 \hat{\imath}+(b+1) \hat{\jmath}+3 \hat{k} \\ & \overrightarrow{A C}=\vec{c}-\vec{a}=4 \hat{\imath}+12 \hat{\jmath}+4 \hat{k} \end{aligned} \] As \(\displaystyle A, B, C\) are collinear \[\begin{gathered} \frac{3}{4}=\frac{b+1}{12}=\frac{3}{4} \\ \Rightarrow b=8 \end{gathered} \] \[\begin{aligned} |\overrightarrow{A C}|= & \sqrt{16+144+16} \\ & =\sqrt{176}=4 \sqrt{11} \end{aligned} \] Here, \(\displaystyle B\) divides \(\displaystyle A C\) in the ratio $\displaystyle 3$ : $\displaystyle 1$