The sum of all values of $\alpha$, for which the points whose position vectors are $$\hat{i}-2 \hat{j}+3 \hat{k}, 2 \hat{i}-3 \hat{j}+4 \hat{k},(\alpha+1) \hat{i}+2 \hat{k}$$ and $9 \hat{i}+(\alpha-8) \hat{j}+6 \hat{k}$ are coplanar, is equal to :

Let $\overrightarrow{O A}=\hat{\mathbf{i}}-2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}$ <br/><br/>$$ \begin{aligned} & \overrightarrow{O B}=2 \hat{\mathbf{i}}-3 \hat{\mathbf{j}}+4 \hat{\mathbf{k}} \\\\ & \overrightarrow{O C}=(a+1) \hat{\mathbf{i}}+2 \hat{\mathbf{k}} \end{aligned} $$ <br/><br/>and $ \overrightarrow{O D}=9 \hat{\mathbf{i}}+(a-8) \hat{\mathbf{j}}+6 \hat{\mathbf{k}}$ <br/><br/>$$ \begin{array}{ll} &\therefore \overrightarrow{A B}=\hat{\mathbf{i}}-\hat{\mathbf{j}}+\hat{\mathbf{k}} \\\\ & \overrightarrow{A C}=a \hat{\mathbf{i}}+2 \hat{\mathbf{j}}-\hat{\mathbf{k}} \\\\ &\text { and } \overrightarrow{A D}=8 \hat{\mathbf{i}}+(a-6) \hat{\mathbf{j}}+3 \hat{\mathbf{k}} \end{array} $$ <br/><br/>Since, given point are coplanar <br/><br/>$$ \begin{aligned} & \therefore \quad[\overrightarrow{A B}, \overrightarrow{A C}, \overrightarrow{A D}]=0 \\\\ & \Rightarrow\left|\begin{array}{ccr} 1 & -1 & 1 \\ a & 2 & -1 \\ 8 & a-6 & 3 \end{array}\right|=0 \\\\ & \Rightarrow 1(6+a-6)+1(3 a+8)+1\left(a^2-6 a-16\right)=0 \\\\ & \Rightarrow a+3 a+8+a^2-6 a-16=0 \Rightarrow a^2-2 a-8=0 \\\\ & \Rightarrow a^2-4 a+2 a-8=0 \Rightarrow a(a-4)+2(a-4)=0 \\\\ & \Rightarrow(a-4)(a+2)=0 \Rightarrow a=4,-2 \\\\ & \therefore \text { Sum of all values of } a=4-2=2 \end{aligned} $$