Definitions & notation

Vectors represent quantities that have both direction and magnitude. Geometrically, a vector is a directed line segment (see Fig. 9.1) whose length is the magnitude and the arrow indicates the direction. Two vectors are the same if they have the same direction and magnitude; i.e. if we take a vector and simply translate it to a new position, we end up with the same vector we started with. The origin of the vector is the point at which it starts; e.g. for the vector in Fig. 9.1, this is the point $A$. Further, a vector is bound if it has a specified origin and it is free if there is no specified origin.

Figure 9.1: A representation of the vector $\overline{A B}$.

In typed notes, vectors are denoted using boldface as in $\boldsymbol{a}$. When writing by hand, vectors are denoted by $\underline{a}$ or $\vec{a}$. To denote the line segment from point $A$ to $B$, the $\overrightarrow{A B}$ or $\overline{A B}$ notation is commonly used. A representation of the vector $\boldsymbol{v}=\left(a_{1}, a_{2}\right)$ in twodimensional (2D) space is any directed line segment $\overrightarrow{A B}$ from $A$ with coordinates $(x, y)$ to $B$ with coordinates $\left(x+a_{1}, y+a_{2}\right)$. In some textbooks, the vector $\boldsymbol{v}=\left(a_{1}, a_{2}\right)$ may also be represented with angle brackets, as $\boldsymbol{v}=<a_{1}, a_{2}>$. In what follows, we give some definitions on the basics of vectors.

Scalars and vectors

A scalar is a quantity that has magnitude only while a vector is a quantity that has both magnitude and direction. Examples of scalars are speed, height, mass, length. Examples of vectors are velocity, force, torque.

Magnitude

The magnitude or length of the vector $\boldsymbol{v}=\left(a_{1}, a_{2}, a_{3}\right)$ is given by,

This is referred to as the Euclidean norm and often also denoted by $\|\boldsymbol{v}\|$.

Unit vector

Any vector with magnitude 1 , i.e. $|\boldsymbol{v}|=1$ is called a unit vector and usually denoted by $\hat{\boldsymbol{v}}$. This is computed by

In this section we mostly used two-dimensional and three-dimensional vectors. Note however that we are not restricted to $2 \mathrm{D}$ and 3D spaces only. In general, we can have an $n$-dimensional vector, $\boldsymbol{v}=\left(a_{1}, a_{2}, \cdots, a_{n}\right)$ where $a_{i}(i=1,2, \ldots, n)$ represent the vector components.

Standard basis vectors

The standard basis vectors for a Euclidean space is the set of linearly independent vectors which are defined with respect to the Cartesian coordinate system, i.e. the axes $x, y$, and $z$ in 3D.

In $3 \mathrm{D}$ space these are given by

Every vector $\boldsymbol{v}$ in $3 \mathrm{D}$ is a linear combination of the standard basis vectors $\mathbf{i}, \mathbf{j}$, and $\mathbf{k}$. Note that standard basis vectors are also unit vectors.