Math 5270
Transformational Geometry
Day 11
Summer 13
Day 11, Summer 13
Math 5270 Transformational Geometry
1/22
Produced withbisector,
a Trial Version
Perpendicular
take 3of PDF Annotator - www.PDFAnno
Day 11, Summer 13
Math 5270 Transformational Geometry
2/22
MyProduced
mistakewith a Trial Version of PDF Annotator - www.PDFAnno
Day 11, Summer 13
Math 5270 Transformational Geometry
3/22
Produced with a Trial Version of PDF Annotator - www.PDFAnno
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withof
a Trial
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Finally,
matrix
an isometry?
We have a matrix
A=
a c
b d
and we’re wondering when it’s a matrix of an isometry T .
necessary:
sufficient:
Day 11, Summer 13
Math 5270 Transformational Geometry
4/22
Invertible matrices
Give a general way of determining if a transformation is invertible.
Can you find the the matrix for the inverse of an invertible
transformation?
Day 11, Summer 13
Math 5270 Transformational Geometry
5/22
Produced
withway
a Trial
of PDFif Annotator
- www.PDFAnno
Give
a general
of Version
determining
the matrix
is
invertible.
Day 11, Summer 13
Math 5270 Transformational Geometry
6/22
Produced
with
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of PDF
Annotator
www.PDFAnno
Can
you find
thea Trial
the matrix
the inverse
of -an
invertible
transformation?
Day 11, Summer 13
Math 5270 Transformational Geometry
7/22
Produced with a Trial Version of PDF Annotator - www.PDFAnno
Day 11, Summer 13
Math 5270 Transformational Geometry
8/22
Produced
a Trial Version of PDF Annotator - www.PDFAnno
What
is thiswith
thing?
Find the area of the parallelogram determined by the given two
vectors.
Day 11, Summer 13
Math 5270 Transformational Geometry
9/22
Produced
withofa the
Trialparallelogram
Version of PDFdetermined
Annotator - by
www.PDFAnno
Find
the area
the given
two vectors.
P.S. Why is that shaded?
Day 11, Summer 13
Math 5270 Transformational Geometry
10/22
Produced
a Trial
of PDF Annotator - www.PDFAnno
Why
is thiswith
thing
not Version
symmetric?
What happens if I find v1 u2 − v2 u1 ?
Day 11, Summer 13
Math 5270 Transformational Geometry
11/22
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with
a Trial Version of PDF Annotator - www.PDFAnno
New
product
[uv]
Properties of this product?
Day 11, Summer 13
Math 5270 Transformational Geometry
12/22
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with
a Trial Version of PDF Annotator - www.PDFAnno
New
product
[uv]
Two vectors u and v determine a number, [uv], called outer
product of u and v, which satisfies the following:
OP1 [uv] = −[uv]
OP2 [(au)v] = a[uv], [u(v + w)] = [uv] + [uw]
OP3 [uv] = 0 when, and only when, u and v are dependent.
Day 11, Summer 13
Math 5270 Transformational Geometry
13/22
Produced
a Trial
of PDF Annotator - www.PDFAnno
Let’s
check with
couple
of Version
favorites:
Find:
1
[(u + v)(u − v)]
2
[(u + v)(u + v)]
Day 11, Summer 13
Math 5270 Transformational Geometry
14/22
a Trial Version
of PDF
- www.PDFAnno
Is Produced
it enoughwith
to require
”[uv] 6=
0 forAnnotator
at least one
pair of
vectors” instead of OP3?
Day 11, Summer 13
Math 5270 Transformational Geometry
15/22
Produced
with a Trial
Version
of a
PDF
Annotator - www.PDFAnno
And
here I thought
the
area of
parallelogram
was base
times height...
Day 11, Summer 13
Math 5270 Transformational Geometry
16/22
Produced
with
a Trial
Version of PDF Annotator - www.PDFAnno
But,
hold on
one
second!
What does [uv] have to do with matrices?
Day 11, Summer 13
Math 5270 Transformational Geometry
17/22
Produced
with
a Trial
Version of PDF Annotator - www.PDFAnno
But,
hold on
one
second!
What does [uv] have to do with matrices?
Remember, matrix
A=
a b
c d
is invertible if and only if ad − bc 6= 0. We call that number we
define
ad − bc
a determinant of A, det A.
Day 11, Summer 13
Math 5270 Transformational Geometry
17/22
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Version of PDF Annotator - www.PDFAnno
What
does with
det Aa Trial
tell me?
Day 11, Summer 13
Math 5270 Transformational Geometry
18/22
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withsay
a Trial
Version
of PDF Annotator
www.PDFAnno
What
can you
about
the determinant
of a- matrix
that
represents an isometry?
1
general
2
reflection
3
rotation
Day 11, Summer 13
Math 5270 Transformational Geometry
19/22
Algebraically determine whether the rotation from (2, 8) to
(−3, −1) is clockwise or counterclockwise
Day 11, Summer 13
Math 5270 Transformational Geometry
20/22
Moment alone
Take a moment to respond to these two question on your own on a
provided notecard.
Explain how you can find a matrix for the reflection about the
line ax + by = c
What was the idea behind the vector proof of concurrency of
medians?
Day 11, Summer 13
Math 5270 Transformational Geometry
21/22
The ending of the journey
Day 11, Summer 13
Math 5270 Transformational Geometry
22/22