If ` vec r=x hat i+y hat j+z hat i ,\ ` find the value of `( vec rxx hat i)dot( vec rxx hat j)+x ydot`

Explanation

To solve the given problem step by step, we need to find the value of the expression \( (\vec{r} \times \hat{i}) \cdot (\vec{r} \times \hat{j}) + xy \), where \( \vec{r} = x \hat{i} + y \hat{j} + z \hat{k} \).

### Step 1: Calculate \( \vec{r} \times \hat{i} \)

We will set up the determinant for the cross product:

\[
\vec{r} \times \hat{i} = \begin{vmatrix}
...