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{SECT 0 {PARA 256 "" 0 "" {TEXT -1 33 "Projections onto Lines and Plan
es" }}{PARA 257 "" 0 "" {TEXT -1 9 "Jim Herod" }}{PARA 258 "" 0 ""
{TEXT -1 21 "School of Mathematics" }}{PARA 259 "" 0 "" {TEXT -1 12 "G
eorgia Tech" }}{PARA 260 "" 0 "" {TEXT -1 21 "herod@math.gatech.edu" }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 " Giv
en a line or a plane through the origin in " }{XPPEDIT 18 0 "R^3" "*$%
\"RG\"\"$" }{TEXT -1 32 ", there are many projections of " }{XPPEDIT
18 0 "R^3" "*$%\"RG\"\"$" }{TEXT -1 62 " onto the line or onto the pla
ne. Abstractly, a projection on " }{XPPEDIT 18 0 "R^3" "*$%\"RG\"\"$"
}{TEXT -1 40 "is defined to be a linear function P on " }{XPPEDIT 18
0 "R^3" "*$%\"RG\"\"$" }{TEXT -1 29 " which has the property that " }}
{PARA 0 "" 0 "" {TEXT -1 45 " \+
" }{XPPEDIT 18 0 "P^2=P" "/*$%\"PG\"\"#F$" }{TEXT -1 1 "." }}
{PARA 0 "" 0 "" {TEXT -1 63 "In this worksheet we will consider only o
ne type projection on " }{XPPEDIT 18 0 "R^3" "*$%\"RG\"\"$" }{TEXT -1
41 " -- namely, we present a formula for the " }{TEXT 256 24 "closest \+
point projection" }{TEXT -1 5 ". In " }{XPPEDIT 18 0 "R^3" "*$%\"RG\"
\"$" }{TEXT -1 40 ", a closest point projection will be an " }{TEXT
258 21 "orthogonal projection" }{TEXT -1 173 ", in the sense that if [
x,y,z] is projected onto a line or a plane, then the line segment from
[x,y,z] to the line or projection will be perpendicular to the line, \+
or plane." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 3 "" 0 ""
{TEXT -1 33 "Orthogonal Projection Onto a Line" }}{PARA 0 "" 0 ""
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 " Consider the line "
}{XPPEDIT 18 0 "L[1]" "&%\"LG6#\"\"\"" }{TEXT -1 134 " containing the \+
point [0,0,0] and the point [a, b, c]. The projection of any point [x,
y, z] onto this line can be created as follows:" }}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 19 "A:=vector([a,b,c]);" }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 36 "P1:=v-> dotprod(v,A)/dotprod(A,A)*A;" }}}{PARA 0 ""
0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 7 "Example" }}
{PARA 0 "" 0 "" {TEXT -1 92 " We take [a,b,c] to be [1,2,3] give t
he dotproduct of [1,1,1,] onto the associated line." }}{EXCHG {PARA 0
"> " 0 "" {MPLTEXT 1 0 15 "a:=1;b:=2;c:=3;" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 22 "P1([1,1,1]);\nevalm(\");" }}}{PARA 0 "" 0 "" {TEXT
-1 39 " We verify that, for this example, " }{XPPEDIT 18 0 "P^2=P
" "/*$%\"PG\"\"#F$" }{TEXT -1 67 " by projecting the previous answer a
nd seeing that this point is a " }{TEXT 257 11 "fixed point" }{TEXT
-1 22 " under this mapping P." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1
0 26 "P1(P1([1,1,1]));\nevalm(\");" }}}{PARA 0 "" 0 "" {TEXT -1 53 " \+
We verify that the line segment from [1,1,1] to " }{XPPEDIT 18 0 "P
[1]" "&%\"PG6#\"\"\"" }{TEXT -1 37 "(1,1,1) is perpendicular to the li
ne " }{XPPEDIT 18 0 "L[1]" "&%\"LG6#\"\"\"" }{TEXT -1 1 "." }}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "dotprod(P1([1,1,1])-[1,1,1],[a,b,c]
);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 34 "Orthogonal Proj
ection Onto a Plane" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 240 "A plane is determined \+
by three points. We take the three points to be \{0, 0, 0\}, \{a, b, c
\}, and \{r, s, t\}. From these three points, we make two vectors -- [
a, b, c] and [r, s, t]. Thus, we make up the projection function with \+
domain all of " }{XPPEDIT 18 0 "R^3" "*$%\"RG\"\"$" }{TEXT -1 61 " and
with range in the plane determined by these two vectors." }}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "restart:\nwith(linalg):" }}}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "A:=vector([a,b,c]);\nR:=vector([r,s
,t]);" }}}{PARA 0 "" 0 "" {TEXT -1 206 "To define the projection, we n
eed two orthogonal vectors in the plane onto which we are making the p
rojection. Thus, we find a vector B which is orthogonal to A and which
is a linear combination of A and R." }}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 33 "B:=R-dotprod(R,A)/dotprod(A,A)*A;" }}}{PARA 0 "" 0 "
" {TEXT -1 38 "We verify that A and B are orthogonal." }}{EXCHG {PARA
0 "> " 0 "" {MPLTEXT 1 0 23 "simplify(dotprod(A,B));" }}}{PARA 0 "" 0
"" {TEXT -1 46 "Now, we can make the orthogonal projection of " }
{XPPEDIT 18 0 "R^3" "*$%\"RG\"\"$" }{TEXT -1 56 " onto the plane by us
ing the orthogonal vectors A and B." }}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 65 "P2:=v->dotprod(v,A)/dotprod(A,A)*A + dotprod(v,B)/dot
prod(B,B)*B;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1
{PARA 4 "" 0 "" {TEXT -1 7 "Example" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT -1 92 " For the purposes of this example, w
e take the plane through the origin to be defined by" }}{PARA 0 "" 0 "
" {TEXT -1 59 " x + 2 y + 3 z \+
= 0." }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 259 5 "first" }{TEXT
-1 127 " job will be to choose the two vectors in this plane. Choose a
ny two points that satisfy the equation. We choose the following." }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "A:=vector([2,-1,0]);\nR:=vec
tor([3,0,-1]);" }}}{PARA 0 "" 0 "" {TEXT 260 6 "Second" }{TEXT -1 89 "
, we find a vector B that is orthogonal to A and that is a linear comb
ination of R and A." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "B:=ev
alm(R-dotprod(R,A)/dotprod(A,A)*A);" }}}{PARA 0 "" 0 "" {TEXT 261 5 "T
hird" }{TEXT -1 88 ", we compute the projection of [1,1,1] onto this p
lane using the formula P2 found above." }}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 12 "P2([1,1,1]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1
0 9 "evalm(\");" }}}{PARA 0 "" 0 "" {TEXT -1 23 "As a check we see tha
t " }{XPPEDIT 18 0 "P^2=P" "/*$%\"PG\"\"#F$" }{TEXT -1 1 "," }}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "P2(P2([1,1,1]));\nevalm(\");" }}}
{PARA 0 "" 0 "" {TEXT -1 141 "Also, we see that the line segment from \+
[1,1,1] to P2([1,1,1]) is orthogonal to the line segment in the plane \+
from P2([1,1,,1]) to [0, 0, 0]." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT
1 0 41 "dotprod(P2([1,1,1])-[1,1,1],P2([1,1,1]));" }}}}}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 24 "Exercise for th
e student" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1
34 "Give the orthogonal projection of " }{XPPEDIT 18 0 "R^3" "*$%\"RG
\"\"$" }{TEXT -1 89 " onto the plane x - y + 2 z = 0. Find the point i
n this plane that is closest to [1,1,1]." }}}{PARA 0 "" 0 "" {TEXT -1
0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "0 0" 33 }
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