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?
Best Answer
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.