Vector

i find it rly ironic that i learned some of this in 10th grade (over quarantine) then forgot it all 😭

Related: Linear Combinations, Span and Basis Vectors

What is a vector?

There are three distinct but related ideas:

• 1. The Physics Student’s Perspective
  • Vectors are arrows pointing in space
  • What defines a vector is it’s length and direction
  • Vectors in a plane are two-dimensional, vectors irl are three-dimensional
• 3. The Computer Science Student’s Perspective
  • Vectors are ordered lists of numbers, like this: $\left[\begin{array}{c}2\\ 1\end{array}\right]$ (looks like [2, 1] in code :3)
  • If I’m making a program about houses, and the only numbers I care about are square footage and price, I could model that with a pair of numbers (a vector (I mean I’d call it a tuple but whatevs (ORDER MATTERS)))
• 2. The Mathematician’s Perspective
  • “The mathematician seeks to generalize both of these views.”
  • “A vector can be anything where there’s a sensible notion of adding two vectors, and multiplying a vector by a number.”
  • This will be ignored until later.

The co-ordinates of a vector are a pair of numbers that basically give instructions for how to get from the tail of that vector (at the origin in most cases :3), to its tip!

• The first number tells you how far to travel on the $x$ axis
  • Positive numbers indicate rightward motion, negative numbers indicate leftward motion
• The second number tells you how far to ravel on the $y$ axis
  • Positive numbers indicate upward motion, negative numbers indicate downward motion
• In 3D, you add a third axis called the $z$ axis, and that’s the third number

To distinguish vectors from points, we write numbers vertically with square brackets around them.

Vector Addition & Multiplication by Numbers

Addition

To add two vectors, move the second so that it’s tail sits at the tip of the first. Then draw a line from the tail of the first vector to the tip of the second, and that’s your new vector!

Formula: $\left[\begin{array}{c}{x}_1\\ y_1\end{array}\right]+\left[\begin{array}{c}{x}_2\\ y_2\end{array}\right]=\left[\begin{array}{c}{x}_1+x_2\\ y_1+y_2\end{array}\right]$

Example: $\vec{v}=\left[\begin{array}{c}1\\ 2\end{array}\right]\vec{w}=\left[\begin{array}{c}3\\ -1\end{array}\right]\vec{v}+\vec{w}=\left[\begin{array}{c}4\\ 1\end{array}\right]$

Multiplication by Number

Multiply each number in the vector by the number you’re multiplying it by. This isn’t rocket science bsfr:

Formula $n\cdot \left[\begin{array}{c}x\\ y\end{array}\right]=\left[\begin{array}{c}nx\\ ny\end{array}\right]$

This is called scaling. The number you’re scaling a vector by is a scalar.