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Use the welghted Euclidean inner productHnere(U,r Uz) and(Vir Vz} to find Iisll; Ihere ' = (-16).MlelIShow MInt220u11 50172...

Question

Use the welghted Euclidean inner productHnere(U,r Uz) and(Vir Vz} to find Iisll; Ihere ' = (-16).MlelIShow MInt220u11 50172

Use the welghted Euclidean inner product Hnere (U,r Uz) and (Vir Vz} to find Iisll; Ihere ' = (-16). MlelI Show MInt 220u11 50172



Answers

Find $\|\mathbf{u}\|$ and $d(\mathbf{u}, \mathbf{v})$ relative to the weighted Euclidean inner product $\langle\mathbf{u}, \mathbf{v}\rangle=2 u_{1} v_{1}+3 u_{2} v_{2}$ on $R^{2}$. $$\mathbf{u}=(-3,2) \text { and } \mathbf{v}=(1,7)$$

Yeah, the norm of you is just the square roots of the inner product of um you and according to the beat heat you pleading inner product vehicles these and this is the square root of two plus 12, which equals the square root out 14. Next we compute the vector v minus you, which of those three and three. So the distance between you envy is just the norm of the vector, the minor issue and vertical these. And it is the square root of 18 last 27. The trickle square road out 45 which is three times the square root out far.

According to the formula the inner product of U. And V equals two times one times three plus three times one times two which is six plus six, which equals 12 for part B. We know that KV echoes 96 and the inner protector of K. V. And W. He cozies which is zero miners, 18 vehicles, miners 18. Percy, you plus one equals +43 And hence the inner product of U plus V. And W. He calls these places which equals zero miners night, which is my nurse night. And the norm of V is just the square root of the inner product of V. And Vi, which he calls two times three times three plus three times two times two. Then the square root which echoes the square root of 18 plus 12, which is the square root out 30 for party. They know that they miners, you echoes to one and the distance of U. And V. Is just the norm of the miners. You which he calls the square root of the inner productive V. Miners, you and itself, which goes two times two times two plus three times one times one, which equals the square root of eight plus three, which is the square root of 11. And you miners kv eccles minus eight minus five and the norm of u minus k. V is just a square root of the inner product of u minus K. V. And itself. So there's echoes hand, Bird and 28 plus 75 which equals the square root of 203.

Hello there. So for this exercise we have these two vectors human being. And we need to calculate first the projection of this metro you along the line spine by this vector V. So basically you can represent that graphically. So that's what network A lets me regret this as a peace for part A. We need to calculate the projection of you on the factory that actually have a geometric meeting. And is that if this this vector he is vector V. And over here you have the vector U. The projection He's just. This is going to be the projection you can be and as you can observe it like rejecting the shadow of you on the case of this part overseas. And for the part B a response to finding the tonal component of you to be. That means which part of the battered you is informational to be. And that is obtain it by taking U minus the projection because the projection is actually how which part of you is align with the would be So if we subtract that, that means you mind as a projection of U. And V. Then you have an Ortho no victory U minus the projection of you. And we will give you these vectors here actually will be or thrown out too deep victor me. Okay, so that is basically the geometric meaning of this exercise. So let's calculating the corresponding projections. So the projection of U. And V is fine as in improved of you would be divided this quarter of norm of B. Times of accurate. So the the square of the norm of B is actually the square Actually is equal to 25/25. She's equals to one. So the norm the norm of B as it goes to one and the inner product of you would be As a fine as one left times Here is a minus sign here. So here's -3 plus 24. But just impulse to 21 over. Fuck. So we have all the components. And we can say that the projection confused on V. Is equal 221 over five times what? Five three war Basically that means 21/25 times three. So great is the projection. And now we're disciplined part. We need to find the orthodontic complaint point to the line of spine spanned by the That means taking U minus the projection of you envy. So this is equivalent too hard. Mine is one six minus This vector that we have a very totaling is 21 over 25 three for and this is equal two 1/25 minus 88. Hama 66.

Hello there. So for this exercise we have these two vectors U. And V. And what we need to do is calculate the component of the orthogonal projection of you and be. So that's basically calculating the projection of UNDP. So jimmy geometrically that means that here we span an infinite line that is represented by these dashed lines. And then here you have the back to you. You can project you on. These line is spanned by the by taking an or throwing a line to two to the line defined by b dessert dessert phono. And then you until you reach the tip off you. Okay So that's why you're referring to our soil projection. And this vector defined here until this point here responsible to the projection of you can be. So basically what you're measuring or calculating shit. He's which the components of you that are aligned with the vector. Okay. Which components are aligned with the vector be in the same direction. And for the second part you need to calculate the component orthe optional to that line. We Okay. So which component of you is orthogonal to this line defined by B. So basically if you have that you you have the protection of you and the defined which components of you are aligned with me then if you take you and use abstract here the projection that means that here is you. And then use obstruct the projection of U. And B. Then you are going to move this vector until this position. This is going to be you mine as a projection. You So if you subtract these regions here you're eliminating the component of you that is aligned with the defined by the So basically you and with components that are so to this like so that's why that professional component of you with respect to the line defined by the factory B is defined in this way. You minus the project? Okay, so there's just some geometric intuition that is important. Me too two. So now let's go to the country to the algebra. So the projection get back to you with respect the victor B is defined as the of you with the back will be were the norm of B squared vector B. So here the inner product of you would be use equal 22 plus three which is equal to five. And the inner product and the square of the norm of B. It is close to the inner product. The weight itself is it constitutes? So then we just need to replace all this data before so we obtained that the projection Oh you on the vector B. Does it find us five House B which is equal to five house. Mhm. What? Okay, so this is the component that of you that are aligned. And now let's calculate the or someone a compliment of you with respect to the line defined by being so u minus the projection if you so we haven't really calculated that this vector so you he's just saying this promoted brain Up here is moving to three minus five house. So the or thrown out component of you to the line defined by B is equal to one. Mhm two minus on half. Um So.


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