Let a → = i ^ + j ^ - k ^ and c → = 2 i ^ - 3 j ^ + 2 k ^ . Then the number of vectors b →…

Given, $\overrightarrow{a}=i+j-k$ and $\overrightarrow{c}=2i-3j+2k$

$\overrightarrow{b}\times \overrightarrow{c}=\overrightarrow{a}$ and $\left|\overrightarrow{b}\right|\in \left\{1,2\dots \dots 10\right\}$

$\because \overrightarrow{b}\times \overrightarrow{c}=\overrightarrow{a}$

$\Rightarrow \overrightarrow{a}$ is perpendicular to $\overrightarrow{b}$ as well as $\overrightarrow{a}$ is perpendicular to $\overrightarrow{c}$ , so $\overrightarrow{a}·\overrightarrow{c}=0$ as they are perpendicular.

Now checking $\overrightarrow{a}·\overrightarrow{c}=2-3-2=-3\ne 0$

This $\overrightarrow{b}\times \overrightarrow{c}=\overrightarrow{a}$ is not possible.

So, number of vectors $\overrightarrow{b}=0$