Section 4 - Vectors and Euclidean Spaces Flashcards
how do you find the angle between two vectors?
cos θ = u•v / |u| |v|
if two vectors u&v are “in the same direction” then v=….
v=ɑu for some ɑ∈ℝ
we say w has direction u from v if…
w-v is a multiple of u
2 line segments are // if ….
they have the same direction
state the vector version of Thales’ theorem
if s&v are on a line through O, and t&w are on another line through O, and w-v is // to t-s, then v=ɑs & w=ɑt for some ɑ∈ℝ
prove the vector version of Thales’ theorem
- if w-v is // to t-s, then w-v = ɑ(t-s)
- since v&s are collinear, v=βs and w=ɣt
- then w-v = ɣt-βs = ɑt-ɑs
- so (ɑ-ɣ)t=(β-ɑ)s
- t&s are on different lines so (ɑ-ɣ)=(β-ɑ)=0
- so v=βs=ɑs & w=ɣt=ɑt
- β=ɑ=ɣ
state the pappus thm
Let l₁ and l₂ be two lines. If A,B,C are on l₁ and a,b,c are on l₂, then the points of intersection of: -Ab&Ba -Ac&Ca -Bc&Cb are collinear
also if Ab//Bc & Ba//Cb, then |Oa|/|OA| = |Oc|/|OC|
what is the vectors version of pappus’ thm
if r,w,v,s,t,u lie alternately on two lines through O with u-v//s-r and t-s//v-w, then u-t//w-r
prove the vectors version of pappus’ thm
- since u-v//s-r we have u=ɑs & v=ɑr by Thales’ thm
- similarly s=βw & t=βv
- then u=ɑs=ɑβw & t=βv=ɑβr
- so u-t=ɑβ(w-r) and u-t//w-r
what is the midpoint of the line from u to v?
(u+v)/2
this can be used to show that the diagonals of a //-gram bisect each other
what are the medians of a △?
the lines joining its vertices to the midpt of the opposite side
where do the medians of a △ meet?
at the centroid (the medians are concurrent)
(u+v+w)/3
what are the operations of the inner (dot) product?
u•v = v•u u•(v+w) = u•v + u•w ɑ(u•v) = (ɑu)•v = u•(ɑv) u•u = |u|² u•v = |u||v|cos θ
what is the vector form of the cosine law
|v-u|² = |v|² + |u|² - 2|u||v|cosθ, where |v-u| is the distance from u to v
prove that non-zero vectors u&v are perpendicular iff u•v=0
- know u•v = |u||v|cos θ
- u perp. to v iff θ=ɑ(𝜋/2)
- so cosθ=0
- then u•v=0
what are the altitudes of a △?
perpendiculars from vertex to opposite side (they are concurrent)
what is a vector version of Pythagoras?
|v-u|² = |v|² + |u|²
state 𝙸.22
To construct a triangle out of three straight lines which equal three given straight lines: it is necessary that the sum of any two of the straight lines should be greater than the remaining one.
state the triangle inequality
if ABC is a △, |AB|+|BC| > |AC|
state the Cauchy-Schwartz inequality
|u•v| ≤ |u||v|
this works in n-dimensions
state the three reflections thm for n-dimensions
any isometry of ℝⁿ is a composition of at most n+1 reflections
define 𝑖
𝑖² = -1
rotation by θ about O corresponds to multiplying each z=x+𝑖y by…..
cosθ+𝑖sinθ
this sends (x,y) to (xcosθ-ysinθ , xsinθ+ycosθ)
state the sum of angles identities
cos(θ₁+θ₂) = cosθ₁cosθ₂ - sinθ₂sinθ₁ sin(θ₁+θ₂) = sinθ₁cosθ₂ + cosθ₁sinθ₂
( these are derived from multiplication by cos(θ₁+θ₂)+𝑖sin(θ₁+θ₂) )
e^(𝑖θ) = …?
= cosθ+𝑖sinθ
e^(𝑖𝜋) = …?
= cos𝜋+𝑖sin𝜋 = -1
state De Moivre’s thm
for n∈ℤ⁺, (cosθ+𝑖sinθ)ⁿ = cos(nθ)+𝑖sin(nθ)
