11-5-2017 Lesson Study research outcomes

at the University of Twente

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Part of our eight-year Lesson Study project

ANALYTIC GEOMETRY

the role of (free) vectors

Nellie Verhoef

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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)

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Vector properties: addition

p + q = q + p (commutative)

(p + q) + r = p + (q + r) (associative)

Move in mathematics

Force in physics

or

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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

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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

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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

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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

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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.

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Visualisation of the dot product geometrical

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Visualisation of p · q = q · p geometrical

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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

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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

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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

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Analytic geometry with free vectors

Geometric center

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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|.

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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.

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Vectors and space

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Hypercube?

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Hypercube unfolded

https://www.youtube.com/watch?v=BVo2igbFSPE

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Hypercube in art

SALVADOR DALI

Questions?

My work is done, Tom Coenen will be the

Lesson Study expertat the University of Twente