The value of$\mathbf{i}\cdot(\mathbf{j}\times\mathbf{k})+\mathbf{j}\cdot(\mathbf{i}\times\mathbf{k})+\mathbf{k}\cdot(\mathbf{i}\times\mathbf{j})$is:

Using vector identities: $\mathbf{i}\times\mathbf{j}=\mathbf{k}$ $\mathbf{j}\times\mathbf{k}=\mathbf{i}$ $\mathbf{k}\times\mathbf{i}=\mathbf{j}$ First term $\mathbf{i}\cdot(\mathbf{j}\times\mathbf{k})$ $=\mathbf{i}\cdot\mathbf{i}=1$ Second term $\mathbf{j}\cdot(\mathbf{i}\times\mathbf{k})$ But $\mathbf{i}\times\mathbf{k}=-\mathbf{j}$ So $\mathbf{j}\cdot(-\mathbf{j})=-1$ Third term $\mathbf{k}\cdot(\mathbf{i}\times\mathbf{j})$ $=\mathbf{k}\cdot\mathbf{k}=1$ Adding $1-1+1=1$