notation, magnitude, scaling & addition Vectors notation, magnitude, scaling & addition

What is a vector? something that has direction and length (AKA “magnitude”) an ordered list of numbers 3 2 Like an “arrow” in space Order matters, these are called “components”

2D Vectors

Position vectors

How is vector different from a point? notation: P = (x, y) means ‘location’ example: place: E&T A221 time: 10 AM v = [ 𝑥 𝑦 ] or v = [x y]T means ‘displacement’ example: distance: 20 steps toward elevators duration: length of our class it happens to be 10 AM – 12:30 PM, but it would be 2.5 hours no matter what time it was scheduled to begin. Analogy to time…

What is the relationship between points and vectors? space time point – point location – location = distance time – time = duration point + vector location + distance = location time + duration = time vector + vector distance + distance = distance duration + duration = duration point + point location + location = ??? time + time = ???

Magnitude of a vector ∆𝑥= 𝑥 2 − 𝑥 1 ∆𝑦= 𝑦 2 − 𝑦 1 𝑣 = (∆𝑥) 2 + (∆𝑦) 2

Vector addition 𝑣 𝑥 𝑣 𝑦 + 𝑤 𝑥 𝑤 𝑦 = 𝑣 𝑥 + 𝑤 𝑥 𝑣 𝑦 + 𝑤 𝑦 𝑣 𝑥 𝑣 𝑦 + 𝑤 𝑥 𝑤 𝑦 = 𝑣 𝑥 + 𝑤 𝑥 𝑣 𝑦 + 𝑤 𝑦 3 2 + 1 −1 = ? ? “as the crow flies” “as the crow flies”

Scaling vectors 𝛼∙ 𝑣 𝑥 𝑣 𝑦 = 𝛼𝑣 𝑥 𝛼𝑣 𝑦 “stretch & squish”