Math 5270
Transformational Geometry
Day 10
Summer 13
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Math 5270 Transformational Geometry
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ax?
Day 10, Summer 13
cos 2θ sin 2θ
where θ = tan−1 a
sin 2θ − cos 2θ
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Directions
Through the origin
Linear independence
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Lines
1. Use vectors to describe the line through points (−1, 2) and
(9, −2).
2. Use vectors to describe the line −3x + 4y = 12
3. Use vectors to describe the line −3x + 4y = 8
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Generalized
directions
w has direction u from v (or relative to v) if w − v is a multiple
of u.
We also say that w − v has direction u
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Parallels
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Parallels
Line segments from v to w and from s to t are parallel if they
have the same direction; that is, if w − v = a(t − s) for some real
number a 6= 0.
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Affine
transformations
Linear transformations preserve:
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Affine
transformations
Linear transformations preserve:
straightness
parallelism
origin
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Affine
transformations
Linear transformations preserve:
straightness
parallelism
origin
To remedy this, we allow composition with translations and these
transformations are called affine
f (u) = Mu + c
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Vector
Thales
Theorem
If s and v are on one line through 0, t and w are on another, and
w − v is parallel to t − s, then v = as and w = at for some
number a.
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Diagonals
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Centroid
ofwith
a triangle
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Towards
distance
u
u= 1 ,
u2
Day 10, Summer 13
v
v= 1
v2
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Things
we know
Magnitude of a vector:
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Things
we know
Orthogonal vectors:
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Towards
distance
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Inner product
For two vectors:
u
u= 1 ,
u2
v
v= 1
v2
we define the inner product with:
u · v = u1 v1 + u2 v2
Equivalently:
u · v = |u||v| cos θ
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How
does this
distance?
What is u · u?
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Criterion
forwith
orthogonality
Two vectors:
u=
u1
,
u2
v=
v1
v2
are orthogonal if and only if u · v = 0.
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Concurrency
of aaltitudes
Theorem
In any triangle, the perpendiculars from the vertices to opposite
sides (the altitudes) have a common point.
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Inscribed
angle
diameter
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1
Take 1: Euclid
2
Take 2: Coordinates
3
Take 3: Vectors
better picture on
the next page
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Triangle
inequality,
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1
Take 1: Euclid
2
Take 2: Coordinates
3
Take 3: Vectors
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