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Find the distance between $\mathbf{x}=\left[\begin{array}{c}{10} \\ {-3}\end{array}\right]$ and $\mathbf{y}=\left[\begin{array}{c}{-1} \\ {-5}\end{array}\right]$

5$\sqrt{5}$

Calculus 3

Chapter 6

Orthogonality and Least Square

Section 1

Inner Product, Length, and Orthogonality

Vectors

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Hello. In this video, we're going to compute the distance between the vectors X and Y, where X is given by 10 in the first component. Negative three in the second component. And why is given by negative one in the first component? Negative five in the second component. And so I think it's always good to visualize the vectors we're trying to compute here. So we have. Let's say this first dimension is X one. The second mentions X two like that here, Um and so that means the the point X given by a vector, uh is it can be given by the 0.10 negative three, Which is going to be, ah, somewhere around here. So this will be X and then why is given by the point negative one negative five, which is going to be given by the point around here? Why and then our goal is to compute the distance between these two, which, if you remember from the Pythagorean theorem, can be computed as this distance will call it a B and C see is what we're trying to sell for and C is equal to the screwed of a squared plus B squared. And so we're actually going to apply that directly here by looking at the differences in the components between these two vectors. So, um, let's get back to competing this magnitude. So the difference between two vectors eyes going to be in this case, the distance between X and Y is going to be equal to this. The magnitude of X minus y which, because of the properties of, um, of vector spaces like the one seen like a two dimensional ones in here, uh, X minus Y is equivalent to why minus X and we'll see why in just a second. Uh, and so, by definition, uh, this distance is going to be equal to the square root of X one minus why one squared the the first component of acts in the first component of why and then, um, that that different squared plus X two minus y two squared the second component, X minus the second opponent why scored? And the reason why these two are equivalent is because we could equivalently do why one minus X one squared plus or to money sex two squared. And these quantities air the same because the square ensures the squares inside this car creates ensure that these quantities air Always positive. We'll just focus on this one on the left eso if we plug our numbers. And now the first component of X is given by 10. 1st component of y is given by negative one. So it'll be 10 minus negative, one squared and then the second component of access given by negative three second component wise given my negative five. So we negative three minus negative. Five squared. Okay, which is equal to if we now simplify these, it'll be 10 plus one squared plus, uh, negative three plus five squared, which is equal to 11 squared plus two squared, which is spirit of 1 21 plus four, which is the skirt of 1 25. Or if we take a five out if we take move 25. Uh, we converted is this First we'll do five times 25 which is equal to the square root of five times the square root of 25 which is equal to square root of five times five or just rewritten. Five squared five. And this is the distance between X and Y. So this see value. And this five squared five, um, are one and the same here. Therefore, the distance between X and Y is going to be given by five Squared five. Thank you.

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