Answer
False
Work Step by Step
We are asked whether the vector-valued function
$\mathbf{F}(x, y) = y\,\mathbf{i} + x^2\,\mathbf{j} + xy\,\mathbf{k}$
is a vector field in the $xy$-plane.
Definition
A vector field in the $xy$-plane assigns a vector in the plane to each point $(x, y)$, i.e.,
$\mathbf{F}(x, y) = M(x, y)\,\mathbf{i} + N(x, y)\,\mathbf{j},$
where $M$ and $N$ are functions of $x$ and $y$. Notice that vectors in the $xy$-plane do not have a $\mathbf{k}$-component.
The given function is
$\mathbf{F}(x, y) = y\,\mathbf{i} + x^2\,\mathbf{j} + xy\,\mathbf{k}.$
$i$ and $j$ components lie in the $xy$-plane.
$k$ component points along the $z$-axis. This is not in the $xy$-plane.
\subsection*{Step 3: Conclusion}
Since the vector field has a nonzero $z$-component, it is \textbf{not a vector field in the $xy$-plane}.
False
A vector field in the $xy$-plane must have vectors lying entirely in the $xy$-plane (only $\mathbf{i}$ and $\mathbf{j}$ components). Here, the $\mathbf{k}$-component makes it a 3D vector field, not a 2D $xy$-plane vector field.
