1.1 - Vector and Unit Vector Notation

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Vectors

Vectors are mathematical objects that represent quantities with both magnitude and direction. In this section we present vector notation that will be used throughout the course.

Vector Notation:
  • Vectors are written as bold symbols (e.g., $\mathbf{r}$, $\mathbf{v}$, $\mathbf{F}$). If you are writing vectors by hand, you may prefer to draw an arrow above the symbol (e.g., $\vec{r}$, $\vec{v}$, $\vec{F}$).
  • The magnitude of a vector is written as an unbolded symbol (e.g., $r$, $v$, $F$). You can also indicate the magnitude of a vector by writing the vector inside vertical bars (e.g., $|\mathbf{r}|$, $|\mathbf{v}|$, $|\mathbf{F}|$).
  • The magnitude represents the length or "size" of the vector, without direction.

Unit Vectors

A unit vector is a vector with magnitude equal to 1 that points in a specific direction. Unit vectors are often used to specify directions and coordinate axes.

Unit Vector Notation:
  • Unit vectors are designated by a "hat" above the symbol, e.g. $\hat{r}$ is the unit vector in the direction of $\mathbf{r}$.
  • We pronounce $\hat{r}$ as "r hat".
  • Unit vectors have magnitude equal to 1, i.e. $|\hat{r}| = 1$

Basis Vectors

A basis vector is a vector that points in the direction of a coordinate axis. Usually, a basis vector is a unit vector, and this will always be true in this course. For example, many textbooks use the unit vectors $\hat{i}$, $\hat{j}$, and $\hat{k}$ to represent the Cartesian coordinate axes $x$, $y$, and $z$ respectively. However, other conventions are also widely used, depending on context.

Common Basis Vector Notations for Cartesian Coordinates:
  • $\hat{i}$, $\hat{j}$, $\hat{k}$ notation: Used for unit vectors along the $x$, $y$, and $z$ axes respectively.
  • $\hat{x}$, $\hat{y}$, $\hat{z}$ notation: Equivalent to $\hat{i}$, $\hat{j}$, $\hat{k}$.
  • $\hat{e}_i$ notation: General notation where $i$ can represent any axis or direction
  • Parentheses notation: Vectors can be represented by their components, such as $(x, y, z)$ or $(v_x, v_y, v_z)$.
In this course, we will use $\hat{x}$, $\hat{y}$, and $\hat{z}$ for most purposes, but we reserve the right to use other notations when it is clearer or more convenient for the particular context.

Examples of Basis Vector Notations:

Let's write the position vector $\mathbf{r}$ using the four conventions introduced above:

Here's how to use the $\hat{x}$, $\hat{y}$, $\hat{z}$ notation used in this course:

$$\mathbf{r} = x\hat{x} + y\hat{y} + z\hat{z}.$$

Equivalently, we can use the $\hat{i}$, $\hat{j}$, $\hat{k}$ notation:

$$\mathbf{r} = x\hat{i} + y\hat{j} + z\hat{k}$$

The $\hat{e}_i$ notation is often used for abstract calculations, but it could be written as:

$$\mathbf{r} = x\hat{e}_1 + y\hat{e}_2 + z\hat{e}_3.$$

The advantage of this method is that we can write the vector components as a summation where $r_1=x$, $r_2=y$, and $r_3=z$:

$$\mathbf{r} = \sum_i r_i\hat{e}_i.$$

Finally, we can simply write the vector as an ordered triple. This lets us write down the vector compoentts without specifying the basis vectors.

$$\mathbf{r} = (x, y, z).$$

Care must be taken when using the last two notations since they can represent the vectors in any coordinate system, not just Cartesian coordinates. When using these notations, one figure out what coordinate system is being used from the context.

Orthonormal Basis Vectors

Most if not all the coordinate systems we will use in this course will be orthonormal coordinate systems. An orthonormal coordinate system is a coordinate system in which the basis vectors are all unit vectors and are mutually orthogonal. In other words, the basis vectors are all perpendicular to each other and have magnitude 1. The Cartesian basis vectors $\hat{x}$, $\hat{y}$ and $\hat{z}$ form an orthonormal coordinate system. As we will see later, the cylindrical and spherical basis vectors are also orthonormal.

Defining a Unit Vector from a Given Vector

Given any vector $\mathbf{r}$, you can always define a unit vector $\hat{r}$ (pronounced "r hat") that points in the same direction as $\mathbf{r}$. You simply divide the vector by its magnitude:

$$\hat{r} = \frac{\mathbf{r}}{r}$$

where $r = |\mathbf{r}|$ is the magnitude of the vector $\mathbf{r}$. This operation normalizes the vector, creating a new vector with the same direction but with magnitude equal to 1.

Key Properties:
  • The unit vector $\hat{r}$ has magnitude $|\hat{r}| = 1$
  • The unit vector $\hat{r}$ points in the same direction as $\mathbf{r}$
  • Any vector can be written in the form $\mathbf{r} = r\hat{r}$, where $r$ is the magnitude and $\hat{r}$ is the direction
Example

Find the unit vector that points in the same direction as the vector $\mathbf{v} = 3 \hat{x} + 4 \hat{y}$.

Solution:

First, calculate the vector's magnitude: $v = |\mathbf{v}| = \sqrt{3^2 + 4^2} = 5$. The unit vector in the direction of $\mathbf{v}$ is:

$$\hat{v} = \frac{\mathbf{v}}{v} = \frac{3 \hat{x} + 4 \hat{y}}{5} = \frac{3}{5}\hat{x} + \frac{4}{5}\hat{y}.$$

You can verify that our unit vector has magnitude 1:

$|\hat{v}| = \sqrt{(3/5)^2 + (4/5)^2} = \sqrt{9/25 + 16/25} = \sqrt{25/25} = 1$.