<p>The solution vectors \(\vec{r}\) of the equation \(\vec{r} \times \vec{i} = \vec{j} + \vec{k}\) and \(\vec{r} \times \vec{j} = \vec{k} + \vec{i}\) represent two straight lines which are:</p>
Step-by-Step Solution
Key Concept: Solve the vector cross product equations to find the two lines, then check if they are coplanar using scalar triple product.
Solving \(\vec{r} \times \hat{i} = \hat{j} + \hat{k}\) gives a line. Solving \(\vec{r} \times \hat{j} = \hat{k} + \hat{i}\) gives another line. Computing the direction vectors and checking the conditions for skew lines, we find the lines are non coplanar (skew) and do not intersect.
Correct Answer: b, d