Section 4 - Vectors and Euclidean Spaces

Question 1

how do you find the angle between two vectors?

Answer 1

cos θ = u•v / |u| |v|

Question 2

if two vectors u&v are “in the same direction” then v=….

Answer 2

v=ɑu for some ɑ∈ℝ

Question 3

we say w has direction u from v if…

Answer 3

w-v is a multiple of u

Question 4

2 line segments are // if ….

Answer 4

they have the same direction

Question 5

state the vector version of Thales’ theorem

Answer 5

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

Question 6

prove the vector version of Thales’ theorem

Answer 6

• 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

Question 7

state the pappus thm

Answer 7

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|

Question 8

what is the vectors version of pappus’ thm

Answer 8

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

Question 9

prove the vectors version of pappus’ thm

Answer 9

• 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

Question 10

what is the midpoint of the line from u to v?

Answer 10

(u+v)/2

this can be used to show that the diagonals of a //-gram bisect each other

Question 11

what are the medians of a △?

Answer 11

the lines joining its vertices to the midpt of the opposite side

Question 12

where do the medians of a △ meet?

Answer 12

at the centroid (the medians are concurrent)

(u+v+w)/3

Question 13

what are the operations of the inner (dot) product?

Answer 13

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

Question 14

what is the vector form of the cosine law

Answer 14

|v-u|² = |v|² + |u|² - 2|u||v|cosθ, where |v-u| is the distance from u to v

Question 15

prove that non-zero vectors u&v are perpendicular iff u•v=0

Answer 15

• know u•v = |u||v|cos θ
• u perp. to v iff θ=ɑ(𝜋/2)
• so cosθ=0
• then u•v=0

Question 16

what are the altitudes of a △?

Answer 16

perpendiculars from vertex to opposite side (they are concurrent)

Question 17

what is a vector version of Pythagoras?

Answer 17

|v-u|² = |v|² + |u|²

Question 18

state 𝙸.22

Answer 18

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.

Question 19

state the triangle inequality

Answer 19

if ABC is a △, |AB|+|BC| > |AC|

Question 20

state the Cauchy-Schwartz inequality

Answer 20

|u•v| ≤ |u||v|

this works in n-dimensions

Question 21

state the three reflections thm for n-dimensions

Answer 21

any isometry of ℝⁿ is a composition of at most n+1 reflections

Question 22

define 𝑖

Answer 22

𝑖² = -1

Question 23

rotation by θ about O corresponds to multiplying each z=x+𝑖y by…..

Answer 23

cosθ+𝑖sinθ

this sends (x,y) to (xcosθ-ysinθ , xsinθ+ycosθ)

Question 24

state the sum of angles identities

Answer 24

cos(θ₁+θ₂) = cosθ₁cosθ₂ - sinθ₂sinθ₁
sin(θ₁+θ₂) = sinθ₁cosθ₂ + cosθ₁sinθ₂

( these are derived from multiplication by cos(θ₁+θ₂)+𝑖sin(θ₁+θ₂) )

Question 25

e^(𝑖θ) = …?

Answer 25

= cosθ+𝑖sinθ

Question 26

e^(𝑖𝜋) = …?

Answer 26

= cos𝜋+𝑖sin𝜋 = -1

Question 27

state De Moivre’s thm

Answer 27

for n∈ℤ⁺, (cosθ+𝑖sinθ)ⁿ = cos(nθ)+𝑖sin(nθ)