5

Point) Find the orthogonal projection ofonto the subspace V of R3 spanned byandprojv (v)...

Question

Point) Find the orthogonal projection ofonto the subspace V of R3 spanned byandprojv (v)

point) Find the orthogonal projection of onto the subspace V of R3 spanned by and projv (v)



Answers

(a) find the projection of $u$ onto $\mathbf{v}$, and (b) find the vector component of u orthogonal to $\mathrm{v}$. $$ \mathbf{u}=\langle 2,-3\rangle, \quad \mathbf{v}=\langle 3,2\rangle $$

First we want to find the projection of you want to be. So um we're going to find the vector component of U. Orthogonal to be after that. So we know that. Are you vector in this case is 033 And if we are the v vector it's going to be negative 111 Yeah. Then um the dot product of these is going to be three plus three. So that's six. So we'll copy this down here and we know that the magnitude is going to be route three. The magnitude of the U. S. Route three. So that means it's down here is just in the three which means this whole thing is going to be too. So now we have two times the vector V. Yeah. Just gonna be negative. 2 to 2. So that's our final answer. And then we could take you and subtract it. We get negative too. Um Are positive too? 11 And that would be the the you vector the vector component of the orthogonal to be

Where the fuck I want this. So We're giving two vectors in R. five. Yeah. You won with components 11341. And you too with components 1 to 1 to one time two. We'll have to find a basis for the subspace W. Bar five orthogonal. To that review one of you two. If the vector V. Lies in the subspace, envy has formed X. Y. V. Uh S. E. T. That's a year. And thanks. Then it follows that the inner product of the that you won in the inner product of these if you two is zero, so we get X plus Y plus three, G. Plus four, S. Plus T. Equals zero and X plus two, Y plus Z. Plus four, S plus T equals zero. I want to solve this system to find basis to do this. Also track the church equation from the 2nd 19. New system, X plus Y plus three, Z plus four S plus T equals zero and X minus 60 to I minus wise. Why Z- Cruzi is -2 Z. And four s. Money for us. A hero. Two months to year zero. All equal zero. Uh Belushi's So looking at this system, we see that there are 3° of freedom. So for example take S. And T. The equal zero. And then you take Z to be equal to one. So he's three days of freedom. And from our second equation it follows that Y equals two. Therefore X is people to -5. Carly. He was just like a pipe dream concert. So we have the 0.-52100. To find other vectors basis vectors. Uh Consider taking, Oh his hair grew long immediately. Yeah, Z. And S equal to zero, but T equals one. Create, create steve. Yeah, everyone's fucking well, the worst thing that could happen, Then it follows from our second equation that Y. is also reported zero. And from the first equation straight X. Is equal to negative one. So we get the vector negative 10001 Finally consider to take uh G. Equal to one and then S. And T. Both be towards zero. Right then. I'm sorry. We already did this instead. Let's take s equal to one and Z. And T. The quarter zero. Then again from the second equation get one equals 0. And from our first equation we get X equals negative six. I would actually see more of it or some. Okay candy did you sloughs? Sorry X should be negative for and maybe six. And therefore he gets instead of three bacterias. Were you good job -10001. Uh negative four 0010 And director negative 52 100 You say that there's one a basis for our subspace W. In our five. Hey?

Hi. So for this exercise we have a vector that is perpendicular to the plane. Okay, so we need to find a parametric equation of this plane, this game coined pie. So how to find this? So basically we have here a condition that we should be perpendicular to this whole plane. That means that any any point X. Y. Z. That lies in the plane. Shit will satisfy that. X. Why Z In a product with 40 -5 Will be equal to zero. Okay. And It will be because 20 because this plane passed through the point through the origin. Otherwise, if this plane passed through some point, let's say V. That is a fracture, then they should be equal to be. So we have this condition to any point X. Times this vector will be professional. So we need to fine all the vectors X, Y and Z. That satisfy this condition. So here we have an equation. The question that we obtain is at four x -5 Y. is equal to zero. So we have to free variables. Here's the Wine. Sorry? Here's -5 Z. So we have to free variables. One is why that's going to be T. And Z. Then this case is going to be S. So defining this, we have the X. Is equal to 5/4 Z. So this gives us a relation of course between the the components. So we can just first some. Yeah. So yeah so the generic solution for this equation here is that the vector X. Why Z? But they find the plane is given in terms of T. M. S. Where the Point where it's passing 30. So I wouldn't put anything. And then here the two vectors that passed through. So we have this is equal to 5/4 S. T. S. And we can separate the solution in two pictures, the picture tea time 010 plus S. Times. Fight over four 01 And this will give us the equation of the plane that we need. You can rewrite actually this vector here, you can choose any multiple. So let's, for example, multiply this vector by four, and you obtain these other representation 010 plus S. Five 04. And this also will be perpendicular to the vector V.

Now we want to find the projection of you want to be. So we're going to consider you now being equal to um 104 and the V vector being and 302 So we take the dot product of these. We get three plus eight is 11. Okay? And then we look at the the magnitude, so we get the square root of nine plus four is 13. Route 13. So we have 13 down here and then we can bring our re vector here, But we end up getting is 33/13 And 22/13. That's the projection. And then we can also subtract in order to get the vector component of U. Orthogonal to be, we would end up getting one minus zero minus days and four minus this.


Similar Solved Questions

5 answers
Find f.f '(x) = 4x + 3/x2 + 3ex + sec? X, f(0) = 10f(x)
Find f. f '(x) = 4x + 3/x2 + 3ex + sec? X, f(0) = 10 f(x)...
1 answers
Consider the followingCompute the characteristic polynomial of A_det(A AI)Compute the eigenvalues and bases of the corresponding eigenspaces of A_ (Repeated eigenvalues should be entered repeatedly with the same eigenspaces:)has elgenspace span(smallest A-value)has elgenspace span
Consider the following Compute the characteristic polynomial of A_ det(A AI) Compute the eigenvalues and bases of the corresponding eigenspaces of A_ (Repeated eigenvalues should be entered repeatedly with the same eigenspaces:) has elgenspace span (smallest A-value) has elgenspace span...
5 answers
Eernkmentnarontr tculodrecinzenanl tm74 Kalt idoJGanenet4ndiun130 50 Euno /110 EoosFicalbllntevulo @Va ttaen & [a iattan4gtlg LMdr Lent Maun Ta eeuaen Ia Inta% 4M motfunt Iptea oecanu Eincos 4 neaad ntam Erpl m valn dt peplatcn In ptnaral '[amdd Maane Iale DeoJ paemalet Laenen Wnettd LE ood obLmutr ouakr mam ticlutd WhtaminntLa @eautr
Eernkmen tnarontr tculod recinzenanl tm74 Kalt idoJ Ganenet 4ndiun 130 50 Euno /110 Eoos Ficalbllnt evulo @Va ttaen & [a iattan4gtlg LMdr Lent Maun Ta eeuaen Ia Inta% 4M motfunt Iptea oecanu Eincos 4 neaad ntam Erpl m valn dt peplatcn In ptnaral '[amdd Maane Iale DeoJ paemalet Laenen Wnettd...
5 answers
Calculate the total mass of plate bounded by y= 0 and y= € for 1 < € < 4 (in meters) assuming mass density of 8 (€,y) 9y/kg/mM=(Use symbolic notation and fractions where needed:)cksoakHint
Calculate the total mass of plate bounded by y= 0 and y= € for 1 < € < 4 (in meters) assuming mass density of 8 (€,y) 9y/kg/m M= (Use symbolic notation and fractions where needed:) cksoak Hint...
5 answers
Quesiicn 4pespXT2 TM}Which molcculc I5 saturated tatty acid?
Quesiicn 4 pes p X T2 TM} Which molcculc I5 saturated tatty acid?...
5 answers
Exercise 5.3.7. In the section On reflections; WC saw that simple reflection across the €-axis, which we will denote by rr; could be expressed T(c,y) (2. V). Let be the reflection across the line y Let T be the translation with displacement vector of v K) Show that the function T- 0 rz T is equal t0 ad] find the coorclinate equation for r. [Hint : Show tlat T1 Tr 0 T has the righc set ol' fixed points:|
Exercise 5.3.7. In the section On reflections; WC saw that simple reflection across the €-axis, which we will denote by rr; could be expressed T(c,y) (2. V). Let be the reflection across the line y Let T be the translation with displacement vector of v K) Show that the function T- 0 rz T is eq...
4 answers
Question 25 (4 points) Determine whether the growth is linear or exponential in the context below and answer the question.The population of Athens is increasing at a rate of 450 people per year: If the population is 696 today; what will it be in three years?a) Exponential; 336,153,316 peopleb) Exponential; 66,430.821 peopleLinear; 2046 peopled) Lincar; 1596 people
Question 25 (4 points) Determine whether the growth is linear or exponential in the context below and answer the question. The population of Athens is increasing at a rate of 450 people per year: If the population is 696 today; what will it be in three years? a) Exponential; 336,153,316 people b) Ex...
5 answers
Hov" do You interpret the residue plot? (2 marks)3kMdua alclo Realddlnla noainaodornlon1Odomolar RuujoFigure
Hov" do You interpret the residue plot? (2 marks) 3kMdua alclo Realddlnla noainaodornlon 1 Odomolar Ruujo Figure...
5 answers
Dfdf Find Jx and dyflx,y) = 2x2Je Ox(Type an exact answer; using radicals as needed )Enter your answer in the answer box and then click Check Answer:part remaining
df df Find Jx and dy flx,y) = 2x2 Je Ox (Type an exact answer; using radicals as needed ) Enter your answer in the answer box and then click Check Answer: part remaining...
5 answers
Required information A scout troop is practicing its orienteering skills with map and compass. First they walk due east for 1.20 km. Next, they walk 45.08 west of north for 2.70 km_How far will they have t0 walk?km
Required information A scout troop is practicing its orienteering skills with map and compass. First they walk due east for 1.20 km. Next, they walk 45.08 west of north for 2.70 km_ How far will they have t0 walk? km...
1 answers
Use a graphing utility to graph $f$ and $f^{\prime}$ on the interval $[-2,2] .$ $$ f(x)=x^{2}(x+1)(x-1) $$
Use a graphing utility to graph $f$ and $f^{\prime}$ on the interval $[-2,2] .$ $$ f(x)=x^{2}(x+1)(x-1) $$...
5 answers
What is the smallest possible value of the principal quantum number n for a p electron?
What is the smallest possible value of the principal quantum number n for a p electron?...
2 answers
Given Hamiltonian with matrix representation of#-(2 :%) Determine if the Hamiltonian is hermitian. Determine if the state vector given byis an eigenstate of the Hamiltonian_Find the normalized energy eigenstates and eigenvalues of the HarniltonianFind the matrix which corresponds to the projection operator for one of your eigen- stales_ Calculate the commutator of this projection operator with the Hamiltonian ([PH])- What does this tell You about the operator P?
Given Hamiltonian with matrix representation of #-(2 :%) Determine if the Hamiltonian is hermitian. Determine if the state vector given by is an eigenstate of the Hamiltonian_ Find the normalized energy eigenstates and eigenvalues of the Harniltonian Find the matrix which corresponds to the projecti...
5 answers
Boyle's Law states that when sample of gas compressed at constant temperature, the pressure and volume V satisfy the equation PV = C, where C is constant: Suppose that at certain instant the volume is 500 cm3 the pressure is 80 kPa, and the pressure is increasing at a rate of 20 kPalmin_ At what rate is the volume decreasing at this instant? cm" /min
Boyle's Law states that when sample of gas compressed at constant temperature, the pressure and volume V satisfy the equation PV = C, where C is constant: Suppose that at certain instant the volume is 500 cm3 the pressure is 80 kPa, and the pressure is increasing at a rate of 20 kPalmin_ At wha...
5 answers
Y = 3 cos(Tx + t)
y = 3 cos(Tx + t)...

-- 0.021803--