Line integral of function from $\mathbb{R}^3 \to \mathbb{R}^3$ along a curve from $\mathbb{R} \to \mathbb{R}^2$ ?! Is this a mistake?

Question

This is an exercise from an exam at my university.

Consider the vectorial field $$f(x,y,z) = (3y^4z^2, 12xy^3z^2, 6xy^4z)$$

(a) Show that $f$ is a gradient field.

(b) Calculate all potentials of $f$.

(c) Calculate $\int_{(0,0)}^{(1,1)}fd\alpha$, where $\alpha$ is a piecewise regular line that unites the points $A(0,0)$ and $B(1,1)$

But if $f$ is from $\mathbb{R}^3$, how can it make sense that $\alpha$ unites the points $(0,0)$ and $(1,1)$ which are from $\mathbb{R}^2$? This has to be a mistake, right? Or am I missing something?