Let a = i +2 j +3 k , b =2 i +3 j -5 k and c =3 i - j +λ k be three...

$$\begin{aligned} & \overrightarrow{\mathrm{r}}=\mathrm{k}(\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}}) \\ & \overrightarrow{\mathrm{r}} \cdot \overrightarrow{\mathrm{a}}=3 \\ & \overrightarrow{\mathrm{r}} \cdot \overrightarrow{\mathrm{a}}=\mathrm{k}(\overrightarrow{\mathrm{b}} \cdot \overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{c}} \cdot \overrightarrow{\mathrm{a}}) \\ & 3=\mathrm{k}(2+6-15+3-2+3 \lambda)\end{aligned}$$3=\mathrm{k}(-6+3 \lambda)$...(1)$\overrightarrow{\mathrm{r}}=\mathrm{k}(5 \hat{\mathrm{i}}+2 \hat{\mathrm{j}}-(5-\lambda) \hat{\mathrm{k}}) |\vec{r}|=k \sqrt{25+4+25+\lambda^2-10 \lambda}=1$...(2)$\mathrm{k}=\frac{3}{-6+3 \lambda}=\frac{1}{-2+\lambda} \quad$put in (2)$$\begin{aligned} & 4+\lambda^2-4 \lambda=54+\lambda^2-10 \lambda \\ & 6 \lambda=50 \\ & 3 \lambda=25\end{aligned}$$