As far as definitions go, understand the difference between a vector and a point. A vector can be translated and still be the same vector, whereas a point is fixed. But I would like some clarification on the usage of vectors and the usage of points, because it seems like in many cases they are used interchangeably. For example:
- It has always been my understanding that addition is not defined for
two points. But in this question, two points are being added
together in this equation:$\overline{C}= \lbrace tx+(1-t)x': x\in C, x' \in C', t\in [0,1]\rbrace$
- Sometimes $\mathbb{R}^n$ is used to denote the set of $n$ dimensional vectors, and sometimes it denotes the set of points in $n$ dimensional space.
- In vector calculus, it is often said that a function with multiple inputs takes a vector as an input, but I have rarely seen a function written as $f(\vec{v})$. Even though I understand that what it means to say is, the vector that originates at the origin, to me, it doesn't seem entirely correct to say that the input is a vector without explicitly saying that the vector's tail is at the origin.
Can anyone clarify these points of confusion? Are point and vector interchangeable in these cases? Also, is there a notation for "converting" one to the other? E.g. how to "convert" $\langle{x,y,z}\rangle$ to $(x,y,z)$ or vice versa?
Best Answer
I'll try to convince you that they are geometrically quite obviously different, but when it comes to naming them, the begin to look alike :)
Geometrically you probably already have a good picture of what a point is: it's just the primitive notion of a point you have in geometry. That is, a single dimensionless location in space.
A vector should be thought of as having two qualities: a ray that has direction and magnitude. In basic vector algebra in $\Bbb R^n$, we learn that such a ray can slide all around $\Bbb R^n$, and as long as you aren't changing the direction or the length of the ray, then it is still the same vector.
Now when it comes to naming these two things, they start to look alike! With Cartesian coordinates, points in $\Bbb R^n$ are labeled by their projections to the axes, and that creates a list of real numbers. Similarly, when we go about naming vectors, we have this convention of sliding the vector so that it is being emitted from the origin, and then we check to see what point is on its arrowhead. The vector is named after this point.
So in both cases, a similar list of real numbers is used to identify the object. Since this is the case, it's common to just start referring to any ordered $n$-tuple of things from a field (like $\Bbb R$) as a "vector," even if we aren't thinking of it as a ray in that application.
One example is that of vector fields. Since these are functions of position, the inputs they take are points of $\Bbb R^n$ (which look like an ordered $n$-tuple). The outputs are vectors (which again look like an ordered $n$-tuple), but we are interpreting these as the vectors they represent, slid over from the origin to the point we're at.
You can, of course, really have vector inputs! For instance, the length of a vector in $\Bbb R^n$ creates a function from vectors into $\Bbb R$. Of course, the same function could be reinterpreted as the distance-to-zero function on points of $\Bbb R^n$.
So, the difference is all in how you are interpreting that particular list of numbers.
For #1 in your post, you are probably thinking of it as the line segment between points $x$ and $x'$. The addition that's going on is vector addition though. Drawing the vectors that $x$ and $x'$ represent, you see you have two vectors extending from the orign to these two points. For any two vectors $v,w$, $v-w$ yields the vector which fits between the two tips of $w$ and $v$, and points to the tip of $v$. So, you can see that $x-x'$ has the point $x$ on its tip.
What does the $t$ contribute? If you multiply out $xt+(1-t)x'=x'+t(x-x')$, you can see that the vector $x-x'$ is being scaled by $t$ to something shorter, and then is being concatenated onto the tip of $x'$. The tip of this arrow gives another point on the segment. Ranging over all $t$ between 0 and 1, you get vectors pointing to all points on that segment.