Given points are $A(1,1,-2)$ and $B(1,-3,1)$
Equation of the line $AB$ is $\overrightarrow r=(\hat i+\hat j-2\hat k)+\lambda (4\hat j-3\hat k)$
Let the point be $C$ which is at a unit distance from $A$
Any point on the line $AB$ is given by $ \hat i+(1+4\lambda)\hat j-(2+3\lambda)\hat k$
Let this point be $C$.
Since $C$ is at unit distance from $A$, $ 16\lambda^2+9\lambda^2=1$
$\lambda=\large\frac{1}{5}$
$\therefore $ $C(\hat i+\large\frac{9}{5}$$\hat j-\large\frac{13}{5}$$\hat k)$
$i.e.,$ $ \large\frac{1}{5}$$(5\hat i+9\hat j-13\hat k)$