If (\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}), find ( (\vec{r} \times \hat{i}) \cdot (\vec{r} \times \hat{j}) + xy)

(\vec{r} = x\hat{i} + y\hat{j} + z\hat{k} \implies \vec{r} \times \hat{i} = -y\hat{k} + z\hat{j} \implies \vec{r} \times \hat{j} = x\hat{k} - z\hat{i} \implies (\vec{r} \times \hat{i}) \cdot (\vec{r} \times \hat{j}) = (-y\hat{k} + z\hat{j}) \cdot (x\hat{k} - z\hat{i}) = -xy \implies (\vec{r} \times \hat{i}) \cdot (\vec{r} \times \hat{j}) + xy = 0)