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Question

Let $\mathbf{y}=\left[\begin{array}{l}{2} \ {3}\end{array}ight]$ and $\mathbf{u}=\left[\begin{array}{r}{4} \ {-7}\end{array}ight] .$ Write $\mathbf{y}$ as the sum of two orthogonal vectors, one in Span $\{\mathbf{u}\}$ and one orthogonal to $\mathbf{u} .$

Foster Wisusik · Accepted Answer

Okay, This question wants us to write the vector. Why, as some of vectors perpendicular and parallel to you, So we&#x27;ll start by finding the parallel component. And that&#x27;s the same thing. Is asking What&#x27;s the component of why along you which we get straight from the projection formula. So we&#x27;re projecting Why on to you? So that would be Why don&#x27;t you divided by the magnitude of you squared distributed into you and the dot product would be two times for minus 21. So eight minus 21 and eight minus 21 is just negative 13. So that&#x27;s negative. 13 The magnitude of you squared would be 16 plus 49 giving 65 and then our vector you is just for negative seven And this vector is the same thing as negative 1/5 times four negative seven. When we reduce, so are parallel Vector would just be negative. 4/5 and 7/5. And now, since we know that why is formed of parallel and perpendicular components, all we have to dio to find the perpendicular component is subtract this parallel component from the original factor. So our original vector Y is 23 and are parallel Vector is negative. 4. 57 50. So why perpendicular would be tu minus 4/5 giving us 14 5th and then three minus 7/5 which would be 8/5. And that is our perpendicular vector. So we have our two answers right here.

Ashley Boni · Answer

so first to find the part that is in the span of you, we want to take the orthogonal projection of why onto you. So why hat is our projection And we know this is equal to white out you over you dot you times you. So why don&#x27;t you is 14 plus six? Uh, you dot you is seven times seven is 49 plus one and we want to multiply this coefficient into the vector or seven. So those coefficient comes out to be 20 over 50 which is 2/5 to fist times seven hiss, 14 over five, and then we get to over five on the bottom. So this is the first part. So now when you define, uh, the other part, that is orthogonal. So we take why minus y hat our Why waas to six union. We subtract our 14 5th in 2/5 to get negative 4/5 and 28 5th So therefore we can, right? Why? As a song of 14 over 52 over five, plus negative for over five. 25

Patrick Horrocks · Answer

so we could use the dot product to find out. I thought you got news are orthogonal. And so the dot product formula, written like this is simply first row of the first window seat, but by first rule of the second radio, and then added to the second of the first liked multiplied by the second hour of the second like a bear. So we simply have seven turns for plus negative, too. Times 14. That is the first year of directives to the product of the second round of the votives. And then that equals 28 for seven times for minus 28. And then that because zero. And so we know if all that desirable for agonal if this dot product equals desire, which we have another word for off agonal is just perpendicular so of a video like this. A then a number of actor coming out here be It&#x27;s the perpendicular don&#x27;t injury. Eight. Have a right angle. I&#x27;m not. It&#x27;s another word for which we get when we can&#x27;t collect the DOT product on that equals zero

Ankit Gupta · Answer

they&#x27;re given Victor&#x27;s are you? Which is equal? Do well. Come on. And maybe it is one going on minus group. No, the projection. Uh, you on TV? Free name it. W one is equal to not productive. You envy Divided by the square off the magnitude of B. Now this will be a scale us on a scale. A multiplication with the activity cilia one comma minus two. It is of activity. No. Now the dot product will be four things. One that is full plus two times minus two, which is minus four. A boy. No absolute value or rather magnet. You off we&#x27;ll be here is square root off one plus those square four that is square root off even than the other square here. So this is only five things, V, which is one comma minus two No. Four minus 40 Thanks One which is Zito Ponzio. Careful, they are Marta. When I or in other words, you does not have any projection on will be. Therefore, we can write you as zero comma zero, which is another way of saying that there is no projection on V plus you itself, which is four former

Shaza Hammoud · Answer

And that&#x27;s a question given the following two vectors, and we have to find the victor projection off you onto V and then write you a song off two or three of Victor&#x27;s one off, which is the projection off you on Tau V. So you should be written as, or the composer&#x27;s tow you one plus you to where these two vectors are orthogonal. You want on you too and you want, since it says that one of them should be the projection off you on Tau V. So I&#x27;ll take you want as this projection you on TV and the formula off the production off you on TV and given a right here, which is deducted product over the magnitude off V squared multiplied by the component for off the victor V. So first, let&#x27;s find the Notre product off U and V, which is equal to the first component off you three. Multiply by the first component of the minus two plus the second component off you minus seven, multiplied by second component of the minus six is equal to minus six in US 42 as mine seven multiplied by minus six is 42 on dhe. This is equals toe 30 six. Now let&#x27;s find the magnitude of a vector V. The magnitude off vector V is the square root off the first component squared, which is minus two squared plus minus six squared. So this is four plus 3600 square root. In other words, it&#x27;s 40 100 square roots. Now let&#x27;s substitute the values inside the projection youand V D formula, which is the daughter product. You got a product off you and Fi as given by 36 the magnitude off we as square root off 40. And it&#x27;s also so the part off too so did pour off to will be cancelled with the square root multiplied by the multiplied by the component for Victor V, which is minus two minus six. So this is is by multiplying thesis Keller number 3 36 over 40. Multiplied by minds to and minus six, this becomes minus 1.8 minus 5.4. So this is the component from off you one. Now you, too from this far from this equation you as you want plus you to than you two. It was tow you minus you want you as three minus seven minus You won, which is minus 1.8 minus 5.4. So this is a three plus 1.8, which is 4.8 then minus seven minus minus 5.4. Just minus seven. Plus 5.4 on This is minus minus 1.6. So this is you too, Andi, this is you want. Now the final think is to just write you in terms off you want and you two just substitute devalues off you and you two inside the equation you want is minus 1.8 minus 5.4 plus 4.8 minus 1.6. So this is D answer O r the d a. You as a some off two orthogonal vector&#x27;s you you want. And you too on this is di Father solution off this question

Linh Vu · Answer

it is questionable given everything you goad u minus Kwan far 11 v He could you thus far, Jew. And first of all, we need you want you compute the scale under But you know with the V by family we&#x27;re getting you don&#x27;t fee over the vida fee. Then we get a multitude minus one times minus four plus four times two And then for the V begin to something B minus for square plus juice square Do we get legal agenda for plus 8/16 blessed for And then we get equal chewed the threw off gente Then we consume defied together six on even in before we can truly under five and therefore confined the protection underwent in yuan of a TV coach in the scale under new three times Fiqh there forget Bridget 3/5 times minus for Jew We get equal June when it&#x27;s thrown over 5 605

Problem

Let $\mathbf{y}=\left[\begin{array}{l}{2} \\ {6}\…

Problem 13 Medium Difficulty

Let $\mathbf{y}=\left[\begin{array}{l}{2} \\ {3}\end{array}\right]$ and $\mathbf{u}=\left[\begin{array}{r}{4} \\ {-7}\end{array}\right] .$ Write $\mathbf{y}$ as the sum of two orthogonal vectors, one in Span $\{\mathbf{u}\}$ and one orthogonal to $\mathbf{u} .$

Answer

$\hat{\mathbf{y}}=\left[\begin{array}{c}{-\frac{4}{5}} \\ {\frac{7}{5}}\end{array}\right]$
$\mathbf{y - \hat { \mathbf { y } }}=\left[\begin{array}{c}{\frac{14}{5}} \\ {\frac{8}{5}}\end{array}\right]$

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David C. Lay, Steven R. Lay, Judi J. McDonald

Linear Algebra and Its Applications

Lily A.

Heather Z.

Kristen K.

Samuel H.

Vectors Intro

Vector Basics - Example 1

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$\hat{\mathbf{y}}=\left[\begin{array}{c}{-\frac{4}{5}} \\ {\frac{7}{5}}\end{array}\right]$$\mathbf{y - \hat { \mathbf { y } }}=\left[\begin{array}{c}{\frac{14}{5}} \\ {\frac{8}{5}}\end{array}\right]$

Linear Algebra and Its Applications

Lily A.

Heather Z.

Kristen K.

Samuel H.

Vectors Intro

Vector Basics - Example 1

David C. Lay, Steven R. Lay, Judi J. McDonaldLinear Algebra and Its Applications

Lily A.

Heather Z.

Kristen K.

Samuel H.

$\hat{\mathbf{y}}=\left[\begin{array}{c}{-\frac{4}{5}} \\ {\frac{7}{5}}\end{array}\right]$
$\mathbf{y - \hat { \mathbf { y } }}=\left[\begin{array}{c}{\frac{14}{5}} \\ {\frac{8}{5}}\end{array}\right]$

David C. Lay, Steven R. Lay, Judi J. McDonald

Linear Algebra and Its Applications