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

$$\begin{aligned}& \mathbf{r} \times \mathbf{b}=\mathbf{c} \times \mathbf{b}\\ & \Rightarrow \quad(\mathbf{r}-\mathbf{c}) \times \mathbf{b}=0 \Rightarrow \mathbf{r}-\mathbf{c}+\lambda \mathbf{b} \\ \text{\text{ or } } \quad \mathbf{r}=\mathbf{c}+\lambda \mathbf{b}\\ & \end{aligned}$$Given, \mathbf{r} \cdot \mathbf{a}=0$, taking dot product with a for Eq. (i). Now,$\mathbf{r} \cdot \mathbf{a}=\mathbf{a} \cdot \mathbf{c}+\lambda \mathbf{a} \cdot \mathbf{b}\therefore \quad \lambda=\frac{-\vec{a} \cdot \vec{c}}{\vec{a} \cdot \vec{b}} \quad[\because \vec{r} \cdot \vec{a}=0] \ldots$(i) From Eqs. (i) and (ii), we get$\mathbf{r}=\mathbf{c}-\frac{\mathbf{a} \cdot \mathbf{c}}{\mathbf{a} \cdot \mathbf{b}} \mathbf{b}$Taking dot with$\mathbf{b}$, we get$\mathbf{r} \cdot \mathbf{b}=\mathbf{c} \cdot \mathbf{b}-\frac{\mathbf{a} \cdot \mathbf{c}}{\mathbf{a} \cdot \mathbf{b}}(\mathbf{b} \cdot \mathbf{b})$where,$\left[$$\begin{array}{l}\mathbf{a}=-\hat{\mathbf{i}}-\hat{\mathbf{k}} \\ \mathbf{b}=-\hat{\mathbf{i}}+\hat{\mathbf{j}} \\ \mathbf{c}=\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}\end{array}$$\right]$=(-1+2)-$\frac{(-1-3)}{(1)}$(1+1)=1+8=9