Vector I

This semester, Vectors was my favorite subject. Maybe because it’s easier for me to visualize it.

In this note, I’ll be documenting the different topics and how I reason about them. A vector is a quantity that has magnitude and a direction. Contrast with scalars which have only magnitude.

Vector representation

Vector being defined as magnitute + direction means we can represent them as something like $(12km, 40\deg)$ or $(8m, \pi)$. But that’s one way of representing vectors. Let’s look at some ways of representing vectors:

• Polar coordinates as mentioned earlier is in the format $(r, \theta)$, where $r$ is the magnitude or radius and $\theta$ is the direction/angle. The figure below demonstrates a polar coordinate.

  This is the plot for $(3,\pi/5)$

• Tuple notation is in the format $\langle v_1, v_2, ..., v_n \rangle$ where $v_i$ is a component of the vector $\vec{v}$. This notation allows for expressing vectors in $n$ dimension unlike the polar coordinate which can be used to represent a vector in just 2 dimensions.

  Familiarly, we know vectors in 2 and 3 dimensions. But generalization is the essence of Mathematics. So some examples are $\langle 4, -5 \rangle$, $\langle 0, 92 \rangle$

  Note that vectors are NOT positional coordinates. Understanding this will prevent you from making some mistakes in future

• Matrix notation is the vertical form of the tuple notation:

  $$\begin{bmatrix} v_1\\ v_2\\ .\\ .\\ .\\ v_n \end{bmatrix}$$

• Unit vector notation: Given a two dimensional (2D) plane, we can have two basis vectors $\vec{i}$ and $\vec{j}$ and the scalars $p$ and $q$ such that $p$ and $q$ scales $\vec{i}$ and $\vec{j}$ respectively giving us:

  $$\vec{v} = p\vec{i} + q\vec{j}$$

  On the 2D plane, we can have $\vec{i} = \langle 1, 0 \rangle$ and $\vec{j} = \langle 0, 1 \rangle$. Taking $p$ as $5$ and $q$ as $2$, we end up with the vector:

  $$\begin{align} \vec{v} &= p\vec{i} + q\vec{j}\\ &= 5\vec{i} + 2\vec{j}\\ &= 5\langle 1, 0 \rangle + 2\langle 0, 1 \rangle\\ &= \langle 5, 0 \rangle + \langle 0, 2 \rangle\\ &= \langle 5, 2\rangle\\ \end{align}$$

  The applet below demonstrates this:

  Throughout this article, I’ll be using the unit vector notation.

Vector addition

Vectors can be added. In practical terms, if we went $12km$ north and then branched $5km$ west, we can be able to tell how far we’ve moved from where we started (ie, the origin).