About representation of a point in space and representation of a vector in space. ("Calculus 7th Edition" by James Stewart.)

Question

I am reading "Calculus 7th Edition" by James Stewart.

To locate a point in space, three numbers are required. We represent any point in space by an ordered triple $(a,b,c)$ of real numbers.

The term vector is used by scientists to indicate a quantity (such as displacement or velocity or force) that has both magnitude and direction. A vector is often represented by an arrow or a directed line segment. The length of the arrow represents the magnitude of the vector and the arrow points in the direction of the vector.

For some purposes it's best to introduce a coordinate system and treat vectors algebraically. If we place the initial point of a vector $\mathbf{a}$ at the origin of a rectangular coordinate system, then the terminal point of $\mathbf{a}$ has coordinates of the form $(a_1,a_2)$ or $(a_1,a_2,a_3)$, depending on whether our coordinate system is two- or three-dimensional (see Figure 11).
These coordinates are called the components of $\mathbf{a}$ and we write $$\mathbf{a}=\langle a_1,a_2\rangle\text{ or }\mathbf{a}=\langle a_1,a_2,a_3\rangle$$
We use the notation $\langle a_1,a_2\rangle$ for the ordered pair that refers to a vector so as not to confuse it with the orderd pair $(a_1,a_2)$ that refers to a point in the plane.

The author identified a point with an ordered triple $(a_1,a_2,a_3)$.
The author identified a vector with an ordered triple $\langle a_1,a_2,a_3\rangle$.
$(a_1,a_2,a_3)$ and $\langle a_1,a_2,a_3\rangle$ are both ordered triples.
But the author strictly distinguishes between them.
Why?

Answer 1

One way to see this is to go back to the classical Euclidean plane, where space is just a set of points with certain properties. Given any two diistinct points, there is a unique line through them. There is also the concept of parallel lines. There is no origin in Euclidean space, and there are no numbers attached to each point. This space is also called affine space.

Given two distinct points, suppose you label one as the start point and the other as the end point. Then there is not only a line segment between the two points, you can put an arrow on the line segment pointing from the start to the end point. A line segment with an arrow is called a vector.

At this point, using only the properties of lines and parallel lines, you can define geometrically the following operations: \begin{align*} \text{point} + \text{vector} &= \text{point}\\ \text{point2} - \text{point1} &= \text{vector}\\ \text{vector1} + \text{vector2} &= \text{vector}\\ \text{vector2} - \text{vector1} &= \text{vector}. \end{align*} You can also rescale a vector by a scalar factor. What is notably missing is the sum of two points. This makes no sense geometrically. It also makes no sense to rescale a point.

The space of all vectors is therefore different from the space of points. Note that in this formulation, there is no special point called the origin. There is, however, the concept of a zero vector, which is the arrow that ends at the same point it starts at.

Now we're ready to write everything in terms of numbers known as Cartesian coordinates. I'll assume you know some abstract linear algebra.

If you choose a basis $(\vec b_1,\vec b_2)$ of the vector space, then there is a natural way to label each vector $\vec v$ by an ordered pair of numbers, $$ \vec v = a^1\vec b_1 + a^2\vec b_2. $$ This is what Stewart writes as $\vec v = \langle a^1, a^2\rangle$.

To label each point in affine space by an ordered pair of numbers, you need to choose an origin $O$ in affine space and a basis $(\vec b_1,\vec b_2)$ of the vector space. Then any point $p$ can be written as $$ p = O + x^1\vec b_1 + x^2\vec b_2. $$ Stewart writes this as $p = (x^1,x_2)$.

It is worth noting that at some point Stewart gives up on all this. If you do choose a fixed origin $O$, then any point can be represented by its displacement vector $$ \vec{r} = P - O = x^1\vec b_1 + x^2\vec b_2 = \langle x^1,x^2\rangle. $$ Starting at some point in the book, Stewart uses displacement vectors only.

Answer 2

Indeed from a foundational perspective, vectors and points are "ontologically" identical. For that matter you could regard complex numbers and two-dimensional vectors of real numbers as identical. What is different is how they are used -- in particular, the operations that we define on them. You can multiply complex numbers but not two-dimensional vectors.

So if we allow ourselves to forget about the fact that points and vectors are ontologically equal, then we'll focus on the difference in how they are used. Points specify a point in space. Vectors specify direction and magnitude (at least for our present purposes). Note that if you take your starting point at the origin, then vectors can be identified with points in space. That is to say, one can equivalently think of points in space, or think of starting from the origin and then traveling along a vector to that point -- and the relationship between these points and vectors is bijective.

However, it is not necessary to select the origin. You can select any point, and any vector, and together they determine a terminal point reached from the start, and then traveling along the vector. Physically it's easy to think of this as the displacement vector, although of course this vector could represent any number of other objects like velocity, force, and so on. Everything I just said about vectors here, cannot be said for points.