Vector I

  • vectors

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,40deg)(12km, 40\deg) or (8m,π)(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,θ)(r, \theta), where rr is the magnitude or radius and θ\theta is the direction/angle. The figure below demonstrates a polar coordinate.

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

  • Tuple notation is in the format v1,v2,...,vn\langle v_1, v_2, ..., v_n \rangle where viv_i is a component of the vector v\vec{v}. This notation allows for expressing vectors in nn 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 4,5\langle 4, -5 \rangle, 0,92\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:

    [v1v2...vn]\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 i\vec{i} and j\vec{j} and the scalars pp and qq such that pp and qq scales i\vec{i} and j\vec{j} respectively giving us:

    v=pi+qj\vec{v} = p\vec{i} + q\vec{j}

    On the 2D plane, we can have i=1,0\vec{i} = \langle 1, 0 \rangle and j=0,1\vec{j} = \langle 0, 1 \rangle. Taking pp as 55 and qq as 22, we end up with the vector:

    v=pi+qj=5i+2j=51,0+20,1=5,0+0,2=5,2\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 12km12km north and then branched 5km5km west, we can be able to tell how far we’ve moved from where we started (ie, the origin).