Vector functions. How do I graph $r(t) = \langle t+1, t-5\rangle$ I am trying to understand Vector functions beyond the trivial textbook definition of magnitude and direction, with the same tired example of a plane flying in a cross-current.  Surely, there must be more than this one application of actually using a vector to model something real.


*

*I understand the basic idea of magnitude and direction.   

*I also get the (0,0) component vector concept.  $V=\langle v1,v2\rangle$ 

*I understand parametric functions.  eg: $x=f(t)$ and $y=f(t)$


Parametric:
$r(t) = (f(t), g(t))$ or written as $x=f(t)$ and $y=g(t)$, then I get it.
For each value of $t$, you can have a distinct $x$ and $y$ and you can plot them.
However, what is the idea behind vector functions??
What does that even mean?
What I don't get is:  $r(t) = \langle f(t), g(t)\rangle$
For example, if I had:
$r(t) = \langle t+1, t-5\rangle$
At $t=1$, $r =  \langle 2,-4\rangle$
At $t=2$, $r= \langle 3,-3\rangle$
What does that even mean?
How would you graph that?
At $t=1$, I would have a vector drawn from $(0,0)$ to $(2,-4)$ ?
At $t=2$, I would have a vector drawn from $(0,0)$ to $(3,-3)$ ?
Would you have an infinite number of component vectors coming out of the origin?  This doesn't sound right.   What is that even supposed to be representing?  Does anyone have an actual real world situation I can use to try understanding what an infinite # of vectors can even represent?
 A: Translate that into a "normal" function of one variable:
$$y=t-5\,,\,\,x=t+1\implies y=x-6$$
and all you have to do now is to draw the above line. Mind you, this is far to be this simple in all cases, but sometimes it is.
A: Such functions are graphed in the same manner as parametric functions. For each value of $t$, there is a dot at the point $\langle t+1,t-5\rangle$. The vector from the origin to this dot is imagined because, as you suggest, having an infinite number of vectors starting at the origin and extending outward would be awkward and unhelpful. If the vector function is continuous, or at least mostly so, you will end up with a curve in the plane, or in the case of the example you gave, a straight line.
If you are interested in more engaging examples of the use of vectors, you should turn to classical mechanics in physics. Start with free-body diagrams. If you know some calculus, you could then learn about electromagnetism and fluid mechanics.
A: Vectors are quite useful. Let me try to explain - 
I hit hard is a complete sentence. I hit him hard is another complete sentence. The first sentence doesn't have any importance to many. The second sentence does. The first one is similar to a scalar, while the second one is a vector.
Using parametric form is nothing but writing a vector of say $2$ dimensions in terms of another quantity t with which we can form the original vector by taking various values for t.
Let me try that with an example again. I gave $10$ coins and $5$ coins is a scalar. I gave $10$ coins to personA and $5$ coins to personB is a vector. I gave to personA $5$ coins more than to personB is representing in parametrics.
I hope I was clear. I tried. 
A: I think this is what's bothering you:

If we're actually interested in the location of a point $(f(t),g(t))$ (a point on a line, in this case), why insist on writing it as a vector $\langle f(t),g(t) \rangle$?

The reason why this is done in many books is that they want to write the parametric equation of a straight line as
$$
\langle x(t),y(t) \rangle
= \langle a+tu,b+tv \rangle
= \langle a,b \rangle
+ t \langle u,v \rangle
,
$$
and in order to do the splitting in the last step, they need to use vector algebra.
Personally, I find this rather artificial, since we actually think of the points $(x(t),y(t))$ and $(a,b)$, and it's only the direction $\langle u,v \rangle$ that's "really" a vector. But to write it in a more natural way, one would have to introduce a new algebraic operation, point + vector = point, like this:
$$
(x(t),y(t))
= (a+tu,b+tv)
= (a,b)
+ t \langle u,v \rangle
.
$$
Apparently many authors consider this to be too confusing for students, since it's almost never seen in elementary textbooks. But as a price for avoiding this algebraic operation, you get other types of confusion, like blurring the distinction between points and vectors, and having to introduce an origin $O$ which really has nothing to do with the matter.
(Compare the nice and clean "point plus vector" equation $$P = P_0 + t \mathbf{v}$$ to the artificial-looking "pure vector algebra" equation $$\vec{OP} = \vec{OP_0} + t \mathbf{v},$$
where you need to pick an arbitrary point $O$ at which to "anchor" your vectors.)
