This is an equation I had in a recent assignment, the question was related to the line integral $$\int (x + 5y)\,\mathrm dx + xy\,\mathrm dy$$ and I was asked to find the parametric equation of the line $C$ between $(0,0)$ and $(3,1)$ if we use the parameter $t$ where $0 \leq t \leq 3$.
I used the equation $(1-t)(0,0) + t(3,1)$ to get the equations $x(t)=3t$, and $y(t)=t$. But the correct answer was $x(t)=t$ and $y(t)=t/3$. I have no idea how those values popped up and was wondering if anyone could help.
The expression $(1-t)(0,0)+t(3,1)$ works just fine in the interval $[0,1]$, but, in this case, you are interested in the interval $[0,3]$ insted. So, use$$\left(1-\frac t3\right)(0,0)+\frac t3(3,1)=\left(t,\frac t3\right)$$instead.