Let ( a = i + 2 j + k ) and ( b = 2 i +7 j +3 k . ) Let ( L _ 1 : r...

$$\begin{aligned} & L_1: \overrightarrow{\mathrm{r}}=(-\hat{\mathrm{i}}+2 \hat{\mathrm{j}}+\hat{\mathrm{k}})+\lambda(\hat{\mathrm{i}}+2 \hat{\mathrm{j}}+\hat{\mathrm{k}}) \\ & \Rightarrow \overrightarrow{\mathrm{r}}=(\lambda-1) \hat{\mathrm{i}}+2(\lambda+1) \hat{\mathrm{j}}+(\lambda+1) \hat{\mathrm{k}} \\ & \mathrm{L}_2: \overrightarrow{\mathrm{r}}=(\hat{\mathrm{j}}+\hat{\mathrm{k}})+\mu(2 \hat{\mathrm{i}}+7 \hat{\mathrm{j}}+3 \hat{\mathrm{k}}) \\ & \Rightarrow \overrightarrow{\mathrm{r}}=2 \mu \hat{\mathrm{i}}+(1+7 \mu) \hat{\mathrm{j}}+(1+3 \mu) \hat{\mathrm{k}}\end{aligned}$$ For point of intersection equating respective components $$\begin{aligned} & \Rightarrow \lambda-1=2 \mu \\ & 2(\lambda+1)=1+7 \mu \\ & \lambda+1=1+3 \mu \end{aligned}$$ We get $$\begin{aligned} & \Rightarrow \lambda=3 \text{ and } \mu=1 \\ & \Rightarrow \vec{a}+\vec{b}=3 \hat{i}+9 \hat{j}+4 \hat{k} \end{aligned}$$\mathrm{L}_3: \overrightarrow{\mathrm{r}}=2 \hat{\mathrm{i}}+8 \hat{\mathrm{j}}+4 \hat{\mathrm{k}}+\alpha(3 \hat{\mathrm{i}}+9 \hat{\mathrm{j}}+4 \hat{\mathrm{k}})$For$\alpha=2, \overrightarrow{\mathrm{r}}=8 \hat{\mathrm{i}}+26 \hat{\mathrm{j}}+12 \hat{\mathrm{k}}$