Graph of parametric vector equation The question is asking me to sketch the graph of the parametric vector function $$\vec r(t) = \vec at+\vec bt^2 $$Where $t$ is a real number, $\vec a$ and $\vec b$ are constant non-parallel non-zero vectors in $\mathbb{R^2}$. 
Putting some arbitrary vectors into Wolframalpha shows the general shape of a parabola which kind of makes sense to me but I'm a bit lost as to the general properties of the graph and how the shape of the graph would change with different vectors. Could anyone explain it to me? Thanks!
 A: Our old friend, the parabola $y=x^2$ has vector equation ${\bf r}(t) = t{\bf i}+t^2{\bf j}$, where ${\bf i}$ and ${\bf j}$ are the unit vectors that point in the positive $x$- and $y$-directions, respectively. 
The choice of ${\bf i}$ and ${\bf j}$ is just the choice of the direction and scale of the coordinate axes. Because ${\bf i}$ and ${\bf j}$ are unit length and at right angles, they give the familiar Cartesian coordinate grid.
There are lots of other choices though. I could have my coordinate axes meet at a $45^{\circ}$ angle, and have one of them measured in inches, while the other is measured in centimetres. Any point on the plane would have a unique pair of numbers - coordinates - even though they would be very different to the usual Cartesian coordinates.
The curve ${\bf r}(t) = t{\bf u}+t^2{\bf v}$ is given in terms of a coordinate system where one axis is parallel to ${\bf u}$, and the other is parallel to ${\bf v}$. The units of are given by the lengths of ${\bf u}$ and ${\bf v}$, respectively.
The important thing is that any pair of non-parallel vectors, say $\{{\bf u}, {\bf v}\}$, can be taken on to any other pair of non-parallel vectors $\{{\bf w},{\bf x}\}$ by a non-singular linear transformation.
This means that the curves ${\bf r}(t) = t{\bf u}+t^2{\bf v}$ are all the same as the curve $y=x^2$, up to a linear transformation. In other words, they are all just stretched and rotated versions of $y=x^2$.
