This preview shows page 1. Sign up to view the full content.
Projections on a line
: LinearAlg
Jonathan L.F. King
University of Florida, Gainesville FL 326112082, USA
[email protected]
Webpage
http://www.math.uﬂ.edu/
∼
squash/
20 November, 2006
(at
18:39
)
Problem
Let
P
=
P
30
◦
be the 2
×
2 matrix whose lefthand action
is orthoprojection on the angle=30
◦
line through the
origin. Compute
P
.
Support your answer in three conceptuallydi±erent
ways.
Strategy
Step
S
0
.
A simplifying notation:
Use
c
:= cos(30
◦
) and
s
:= sin(30
◦
).
Step
S
1
.
Use similar triangles to compute
P
·
ˇ
e
1
;
this product gives the ﬁrst column of
P
. Use sim.tri
to compute
P
·
ˇ
e
2
; this equals the second column of
P
.
Step
S
2
.
Checking our result: Each vector
v
on
the 30
◦
line should be
ﬁxed
(not moved) by the pro
jection. So
P
·
ˇ
v
better equal ˇ
v
.
Step
S
3
.
“Variation of parameters”: Generalize
the result of Step1, replacing “30
◦
” by a general an
gle
ω
. Easily
P
0
◦
=
"
1 0
0 0
#
and
P
90
◦
=
"
0 0
0 1
#
.
1
:
Now we will check that our
This is the end of the preview. Sign up
to
access the rest of the document.
This note was uploaded on 01/26/2012 for the course MAC 3472 taught by Professor Jury during the Fall '07 term at University of Florida.
 Fall '07
 JURY
 Calculus, Vectors

Click to edit the document details