fl D ortho IT is Projections g g LY 1.4 If J by then Mr We wish to write ig as a sum of two special where 2 is in the samed tas and This

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1 1.4 Projections g where 2 is in the samed tas and T is ortho This is when easy to Suppose we have two vectors and g n R to We wish to write ig as a sum of two special vectors f J by then Mr a E g LY t g E 5 parallelto f perpendicularto xf.to Yj fl D

2 Q How do we find t Ez when 5 is more complicated R2 N Y an Y N r r fzi N s t L f Zi N N We Know that 1 is parallel to 5 i.e a multiple of F so 2 K for some KER This means that F 2FtZz KxtZzw where Zz is orthogonal to 5 To find k take the dot product of E and J g Ket 2 2 KK TE HM as E tx

3 So g KNK K X we've just shown that Z Since g Ei Definition f and are in R with to then the projection of g onto 5 is projxy fj.gg Y and the perpendicular part is perp J g proj g From this definition it's clear that lily.pro Perpy and ii Pro jg is a multiples n Q2 of Assignment 2 you verify iii Perp y J fxj.t isorthogona tox

4 E What is the projection of onto E what is the perpendicular part f Solution we have that proj y Y x We get and g E 4 10 So proj g 5 25 and perp g j projej L L f Nz A q pj 1 tix E se Ex f and prog'g o and 5 find to ProjeJ perp g 3 0

5 Solution Note that and s proj.j fxixyjx.fi x g x lo ff.o z O Z perp j j projxy f.io f J ff 3 3 For projge note that Hugh 104 and g 10 so projgx ffgffy.fi g TRemar fi oi K The above example shows that in general proj ftp.nojgx That's not too surprising proj J is a multiple of 5 while projg is a multiple of g theorem properties of Projl Perp Let J E be vectors in Rh and let te R E

6 z p.j.mg 3 ProjetProj y Proj Lg These properti are also true for Perp and proj perp are linear Properties 1 and 2 say that functions from R into R we'll return to something in Chapter 3 Proofoft ProjecgiEl TEA fifth x tfi.t x Proj.jtProix E.u Exercise Try a similar approach for 2 Can you explain intuitively why 3 is true

7 Application Minimum Distance e w QUESTON about projections There are Lots of reasons to care about projections One major application projections can be used to find the line ofbestfit for a set of data points x 1 This is the line whose total distance to a set of points is as small se as possible A complete treatment of this topic MATH 235 nstead we'll look at two different applications of projections to minimum distance Want to know more about how to find this line Talk to me after chapter3

8 1 Distance from Point to Line mm what's the distance i e Shortest distance from a point Q to a line 5 15 ttd ter Rzµ P t Q t's the length of the f dotted line that meets n our line at a right angle Rz n the language of projections N this is exactly ftp.pq Fa TtQ P p a The point on the line Proyectot T closest to Q is F i apoint n Ex Find the distance from Q O 2 to the line 4 2 tf1 te R What is the closest point

9 Solution We have that and 111 Z 1 From the above the distance 11 perp H and the closest point is ft Projj Fa So let's find Projjffa and Perp P Note Fa 4 4 E F z z o d Fa 4 so Projet fi Y f 3 Perp P Prog d POT Y The distance is Hperpi The closest point is ft Projj Z z o

10 Exercise Find the distance from Q 1,0 l to the line f tf te R What is the closest point 2 Distance from Point to Plane what's the distance from a point Q to a plane in 1123 with normal vector ri f'd suppose P is any point Q on the plane t p b fi The distance is the length of the dotted line segment that is orthogonal to the plane n terms of projections in A Perp P this is Mill T The point on the plane closest to Q is jpq 1 This is the same as ftp.rpjpq kingt ftp.a b a AA closestpoint

11 E What is the distance from Qf 1,421 to the plane kit Find the point on the plane that is closest to Q Solution We can a point P on the plane get by setting Nz Ks 0 and solving for se Kit N 4 se se 0 So P C 4,0 o is a point on the plane and T J From above we know that the distance is HProjaFall and the closest point is of Projata We have POT E p f f Y f Z 3 Z ri PQ 9 So Projita Finton ri L The distance is HProjn.PEl rq 3

12 The closest point is E Project l t.t Exercise what is the distance from Qtl l l to the plane 2x sea t se 2 Notice anything odd Which point on this plane is closest to Q