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!
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 $y=x^2$.