# Scalars and Vectors

We have already seen the definition of a scalar when working with matrices, however, we will now redefine it so that it makes a little more sense with regards to vectors (which we also define).

Definition: A Scalar is a numerical quantity that has a magnitude but no direction (e.g. $0.5$, $3$, etc…). A Vector is a quantity that has both a magnitude and a direction (e.g. $5$ left, $2$ west, etc…). |

For most of the linear algebra pages we will denote a scalar by a lowercase letter, usually $k$, $m$, or $l$, while we will generally denote a vector with an arrow such as $\vec{u}$ or sometimes bolded characters such that $\mathbf{u}$. We will write $\vec{u} \in \mathbb{R}^n$ if we want to specify that the vector $\vec{u}$ is in Euclidean $n$-space or is in $n$-dimensional space. For example, $\vec{v} \in \mathbb{R}^2$ means that $\vec{v}$ is in 2-dimensional space.

Geometrically, we can say that a vector in $n$-space is comprised of $n$-components which describes the vector itself, that is $\vec{u} = (u_1, u_2, …, u_n)$. Geometrically, a vector in 2-space or 3-space can be represented as a straight line with an arrowhead:

Before we talk about vectors further, we will introduce some important definitions.

Definition: Let $\vec{u} \in \mathbb{R}^n$. Then the Initial point of $\vec{u}$ is the start of the arrow that represents $\vec{u}$ and the Terminal point of $\vec{u}$ is the end of the arrow that represents $\vec{u}$. The Origin is the point whose components are all zero, that is, the point where all the axes of the coordinate system intersect. |

For example, consider the vector that has its initial point at coordinates $P$ and terminal point at coordinates $Q$. We can draw a vector from $P$ to $Q$ and denote this vector $\vec{PQ}$:

Sometimes an initial point and terminal point are not specified, for example the vector $\vec{u} = (2, 4)$. We can thus draw this vector anywhere in the 2-space plane, however it is common to place the vector at the origin.

Therefore, if an initial point and terminal point is not relevant for a vector, then the vector itself can be represented geometrically as a line with an arrowhead or just a point.