Mathematics of "directed" quantities
In many applications, we need to manipulate quantities that contain not only magnitude (how big) but also direction (which way) information. In mathematics, these are captured by the abstract concept of "vectors".
A vectors is a geometric object that has both magnitude and direction. In the plane they can be represented by "directed" line segments. In this lecture, we will learn how to represent and carry out calculations with vectors.
In many applications, we need to manipulate quantities that contain not only magnitude (how big) but also direction (which way) information. Familiar examples of such quantities include force, velocity, and acceleration. In mathematics, these are captured by the abstract concept of "vectors".
In the plane, they can be represented by "directed" line segments. We will learn how to represent and carry out calculations with vectors.
Vectors are represented as line segments with arrowheads.
A "directed" line segment is simply a line segment with an assigned direction. The two end points are called its initial point and terminal point respectively, and we usually draw an arrowhead from the initial point to the terminal point to indicate the direction.
The length of the line segment represent the magnitude of the vector.
In mathematics, it is most common to use lowercase boldface letters for names of vectors (e.g. $\mathbf{v}$ or $\mathbf{u}$). When it is difficult to simulate boldface types, we often use small arrowhead over lowercase letters (e.g. $\vec{v}$ and $\vec{u}$).
In order to be able to carry out computation with vectors, it is also necessary to include a special vector that has zero magnitude --- the "zero vector".
The zero vector is the only vector that has no assigned direction, and it is usually represent by a single point.
It is usually denoted by $\mathbf{0}$ (the boldface 0). Alternatively, we can use the notation $\vec{0}$ if boldface type is difficult to simulate. Whichever notation we use, we must distinguish it from 0 (the number) as they are very different.
Even though we represent vectors by directed line segment, it is important to understand vectors are not "attached" to points.
In other words, in the representation of a vector, the initial and terminal points are not important.
Only its magnitude (length) and direction are important, and you are free to draw it anywhere you like as long as these two properties are preserved.
For two equivalent vectors $\mathbf{v}$ and $\mathbf{u}$, we write $\mathbf{v} = \mathbf{u}$.
In calculations, vectors often interact with numbers (real numbers), so it crucial to distinguish these two classes of objects.
In our context, we also call a real number a scalar to emphasize this important distinction. Essentially, a scalar has a magnitude, but no direction.
For example, the velocity of an object is a vector, but its speed is a scalar.
We usually use regular (non-boldface) lowercase letters for names scalar quantities (but the distinction should always be clear from the context).
For a vector $\mathbf{v}$ in the plane, its magnitude is denoted by $\| \mathbf{v} \|$, and it is a nonnegative real number.
Indeed, the zero vector $\mathbf{0}$ is the only vector with 0 magnitude. All other vectors have positive magnitude.
In certain context, the magnitude of a vector is also known as its norm.
There are infinitely many unit vectors. Indeed, picking any point on the unit circle (centered at the origin), the directed line segment from origin to this point gives us a unit vector. Of course, there are infinitely many such vectors.
When carrying out computation involving vectors in a plane, it is much preferred to work in a coordinate system. There are many coordinate systems. The Cartesian coordinate system is the most straightforward one, and we will work within this system for most of the lectures in this course.
Since a vector can be "placed" anywhere, we can always move it so that its initial point is the origin.
Once in this position, we can use the $x$ and $y$ coordinates of the terminal point to represent this vector.
If a vector has the origin as its initial point and the point $(x,y)$ as its terminal point, then we can use the pair of number $x,y$ to represent it.
We can use a variety of notations \[ (x,y) \quad\text{or}\quad \langle x, y \rangle \quad\text{or}\quad \begin{bmatrix} x \\ y \end{bmatrix} \]
Here, $x$ and $y$ are called components of this vector.
Using Pythagorean Theorem, we can compute the magnitude of a vector from its components easily.
From this formula, we can also see that the magnitude of a vector must be a nonnegative real number.