analytic geometry...analytic geometry with vectors orthocentre choose the origin o, the intersection...

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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

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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

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