Determine which of the vectors $\mathbf{v}_1=\left(\begin{array}{l}1 \\ 1 \\ 0\end{array}\right), \mathbf{v}_2=\left(\begin{array}{r}-2 \\ 2 \\ 2\end{array}\right), \mathbf{v}_3=\left(\begin{array}{r}2 \\ -1 \\ -3\end{array}\right), \mathbf{v}_4=\left(\begin{array}{r}-1 \\ 3 \\ 4\end{array}\right)$, is orthogonal to (a) the line spanned by $\left(\begin{array}{r}1 \\ 3 \\ -2\end{array}\right) ;(b)$ the plane spanned by $\left(\begin{array}{r}1 \\ -1 \\ 1\end{array}\right),\left(\begin{array}{l}2 \\ 1 \\ 1\end{array}\right)$; (c) the plane defined by $x-y-z=0 ;(d)$ the kernel of the matrix $\left(\begin{array}{lll}1 & -1 & -1 \\ 3 & -2 & -4\end{array}\right)$; (e) the image of the matrix $\left(\begin{array}{rr}-3 & 1 \\ 3 & -1 \\ -1 & 0\end{array}\right)$;
(f) the cokernel of the matrix $\left(\begin{array}{rrr}-1 & 0 & 3 \\ 2 & 1 & -2 \\ 3 & 1 & -5\end{array}\right)$.