analytic geometry...analytic geometry with vectors orthocentre choose the origin o, the intersection...
Post on 12-Aug-2020
1 Views
Preview:
TRANSCRIPT
11-5-2017 Lesson Study research outcomes
at the University of Twente
1
Part of our eight-year Lesson Study project
ANALYTIC GEOMETRY
the role of (free) vectors
Nellie Verhoef
11-5-2017Lesson Study research outcomes at the University of Twente 2
History
Euclid (third century bc)
Analytic geometry or coordinate geometryin contrast to
synthetic (axiomatic) geometry which dates back to the ancient Greece
The founders of the analytic geometry are the French mathematicians:
René Descartes (1596-1650)Pierre de Fermat (1601-1665)
11-5-2017Lesson Study research outcomes at the University of Twente 3
Vector properties: addition
p + q = q + p (commutative)
(p + q) + r = p + (q + r) (associative)
Move in mathematics
Force in physics
or
11-5-2017Lesson Study research outcomes at the University of Twente 4
The existence of an element zero 0 without direction,
each p has its opposite -p
1·p = p
λ(p + q) = λp + λq
(λ+μ)p = λp + μp
λ(μp)=(λμ)p
Vector properties: addition and scalar multiplication
11-5-2017Lesson Study research outcomes at the University of Twente 5
The choice of an origin and a coordinate system
Point P(p1,p2) on line l, with coordinates p1and p2
analogue to point P in space with three coordinates
11-5-2017Lesson Study research outcomes at the University of Twente 6
The dot product
An algebraic and a geometric definition:
Analogue in space withthree coordinates
Same definition? Yes, read the last Euclides!
p · q = 0 p ┴ q
11-5-2017Lesson Study research outcomes at the University of Twente 7
The dot product
Analogue in space with three coordinates
The dot product can also be described in terms
of determinants.
11-5-2017Lesson Study research outcomes at the University of Twente
Vector equations of lines
Analogue in space withthree coordinates
11-5-2017Lesson Study research outcomes at the University of Twente 9
The distance from a point P to a line l algebraical
Analogue the distance from a point P to a plane Vin space with three coordinates
11-5-2017Lesson Study research outcomes at the University of Twente 10
In our Lesson Study team we posed the question:
We know how to calculate the dot product,the answer is a number, what does that number mean?
Direction? Length? Angle?
John, a member of our Lesson Study team, tried to find an answer…. based on his beliefs about the Greek vision of
multiplication as an area.
11-5-2017Lesson Study research outcomes at the University of Twente 11
Visualisation of the dot product geometrical
11-5-2017Lesson Study research outcomes at the University of Twente 12
Visualisation of p · q = q · p geometrical
11-5-2017Lesson Study research outcomes at the University of Twente 13
The distance from the origin O to a line l, P є l, geometrical
Line l: (n · x – p) = 0 (n · x) = (n · p)
We see (n · x) = 10,
d(O,l)=√10
11-5-2017Lesson Study research outcomes at the University of Twente 14
The distance from a point to a line l, P € l, geometrical
Line l: (n · x – p)=0 (n · x) = (n · p)
We saw (n · x) = 10,
d(O,l)=√10
We see (n · x) = 20,
(n · x) = -5
11-5-2017Lesson Study research outcomes at the University of Twente 15
Geometry without free vectors
Geometric center
AD, BE and CF are medians.ED bisects AC and BC, ED=½AB and ΔABZ~ΔDEZ => AZ=2ZD and BZ=2ZE.CF intersects AB also with ratio 2:1 => Z is the geometric centre.
The Lesson Study team emphasizes differences
11-5-2017Lesson Study research outcomes at the University of Twente 16
Analytic geometry with free vectors
Geometric center
11-5-2017Lesson Study research outcomes at the University of Twente 17
Geometry without vectors
Orthocentre
AD, BE and CF are altitudes.Double the sides of the triangle – contour lines become perpendicular.Bisectors of ∆GJI cut in one point S(=H) because |SG| = |SI| |SI| = |SJ| |SG| = |SJ|.
11-5-2017Lesson Study research outcomes at the University of Twente 18
Analytic geometry with vectors
Orthocentre
Choose the origin O, the intersection point of the altitudes AD and BE.Three vectors: a, b and c
We know that a (b – c) = 0 and b (c – a) = 0 that meansa c = a b = b c => c (a – b) = 0 or OC AB,O(=H) is an element of the altitude through C.
11-5-2017Lesson Study research outcomes at the University of Twente 19
Vectors and space
11-5-2017Lesson Study research outcomes at the University of Twente 20
Hypercube?
11-5-2017Lesson Study research outcomes at the University of Twente 21
Hypercube unfolded
https://www.youtube.com/watch?v=BVo2igbFSPE
11-5-2017Lesson Study research outcomes at the University of Twente 22
Hypercube in art
SALVADOR DALI
Questions?
My work is done, Tom Coenen will be the
Lesson Study expertat the University of Twente
top related