Determine whether the vectors are coplanar.
$i, i − 2j, 3j + k.$
My first intuition tells me that one must multiply the given points to find out the answer. So that is $a*b*c$ so that gives you zero. $1*1*0+0*-2*3+0*0*1 = 0$. Do you multiply the coordinate points or the variables?
$i$, $j$, and $k$ are not variables, they are the standard unit vectors.
Here's one way to do it: Make an equation of a plane out of two of the vectors and see if the other can solve it. For instance $$\mathbf r(u,v) = (\mathbf i)u + (\mathbf i - 2\mathbf j)v = (u+v)\mathbf i -2\mathbf j$$
Can you find numbers $u$ and $v$ such that $\mathbf r(u,v) = 3\mathbf j+\mathbf k$
Another method you could use to construct a matrix out of these three vectors and find the determinant. If it's zero, then the vectors must be linearly dependent which in this case means they must be coplanar. One matrix you could construct is
$$\begin{bmatrix} 1 & 0 & 0 \\ 1 & -2 & 0 \\ 0 & 3 & 1\end{bmatrix}$$