Subsection2.8.2Representation of Vectors and Vector Notation
Since vectors describe both magnitude and direction, we cannot represent them using simple numbers alone. To communicate mathematical ideas clearly, we need a standardized way to draw and write vectors so that everyone interprets them in exactly the same way. The following activity will help you explore how to visualize and record these movements before we define the formal notation.
On a piece of graph paper, draw a sketch that shows your path from point \(\text{A}\) to \(\text{B}\) and then from \(\text{B}\) to \(\text{C}\text{.}\)
Think of a way on how you can represent the movement from point \(\text{A}\) to \(\text{B}\) using notation, also from point \(\text{B}\) to \(\text{C}\text{.}\)
Suppose you now move in a reverse way from point \(\text{C}\) to point \(\text{A}\) following the same path. How would you represent that movement using notation?
Vector notation is a way of representing quanties that have both magnitude and direction. Vector \(\textbf{PQ}\) can be denoted as \(\underset{\sim}{PQ}\) or \(\overrightarrow{PQ}\) or \(\mathbf{PQ}\text{.}\)The magnitude of vector \(\textbf{PQ}\) is represented as \(|\textbf{PQ}|\text{.}\) In this case, we refer to \(P\) as initial point and \(Q\) as the terminal point.
Additionally, a vector can also be represented using a single small letter,such as \(\mathbf{a}\) or \(\underset{\sim}{\mathbf{a}}\text{.}\) In Figure 2.8.7 below, we can represent a vector from a point \(\text{P}\) to point \(Q\) as \(\overrightarrow{PQ} = \mathbf{a} = \underset{\sim}{\mathbf{a}}\)
Similarly, if the direction of the vector is reversed, from point \(\text{Q}\) to point \(\text{P}\) the vector is represented as \(\overrightarrow{QP} = \mathbf{-a} = -\underset{\sim}{\mathbf{a}}\)
