For any vector r , what is ( r · i )( r × i )+( r · j )( r × j )+(...

$(\vec{r} \cdot \hat{i})(\vec{r} \times \hat{i})+(\vec{r} \cdot \hat{j})(\vec{r} \times \hat{j})+(\vec{r} \cdot \hat{k})(\vec{r} \times \hat{k})$ ...(i) We have, $$\begin{aligned} \vec{r} & =a \hat{i}+b \hat{j}+c \hat{k} \\ (\vec{r} \hat{i}) & =(a \hat{i}+b \hat{j}+c \hat{k}) \cdot \hat{i}=a \\ (\vec{r} \cdot \hat{j}) & =b \\ (\vec{r} \cdot \hat{k}) & =c\end{aligned}$$ and \qquad \vec{r} \times \hat{i}=\left|$\begin{array}{lll} \hat{i} & \hat{j} & \hat{k} \\ a & b & c \\ 1 & 0 & 0 \end{array}$\right|=-\hat{j}(-c)-b \hat{k}=c \hat{j}-b \hat{k}$ Similarly, $\quad $\vec{r}$ \times \hat{j}=-c \hat{i}+a \hat{k}$ and\vec{r} \hat{k}=b \hat{i}-a \hat{j}$ Putting these values in (i) we get\begin{aligned} & a(\hat{j}-b \hat{k})+b(-c \hat{i}+a \hat{k})+c(b \hat{i}-a \hat{j}) \\ &=a c \hat{j}-a b \hat{k}-b c \hat{i}+a b \hat{k}+b c \hat{i}-a c \hat{j}=0 \end{aligned}$$