I encountered this problem in one of the tests: Let $\overrightarrow{a}=\overrightarrow{b}\times \overrightarrow{c}$. If $\overrightarrow{a}=\hat{i}+\hat{j}-\hat{k}$ and $\overrightarrow{c}=\hat{2i}-\hat{3j}+\hat{2k}$. Then total number of $\overrightarrow{b}$ such that $|\overrightarrow{b}|\epsilon \{1,2....10\}$.
My approach was to assume the $\overrightarrow{b}=\alpha\hat{i}+\beta\hat{j}+\gamma\hat{j}$, take the cross product with $\overrightarrow{c}$ (using the determinant) and then compare it with $\overrightarrow{a}$, to get the required value. But the value that I am getting, gives the absolute value of $\overrightarrow{b}<1$, so should the answer be $0$, or is there something else that I am missing?
Here is the image of the question for reference.
The number of possible vectors $\vec{b}$ is zero, since $\vec{a}$ and $\vec{c}$ are not perpendicular. The information about the magnitude $|\vec{b}|$ is irrelevant.
$\vec c \cdot (\vec b \times \vec c)$ is always zero, but $\vec{c} \cdot \vec{a} = +2-3-2=-3$