Scalars and Vectors

# 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.