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Use the inner product axioms and other results of this section to verify the statements in Exercises $15-18$ .$\langle\mathbf{u}, \mathbf{v}\rangle=\frac{1}{4}\|\mathbf{u}+\mathbf{v}\|^{2}-\frac{1}{4}\|\mathbf{u}-\mathbf{v}\|^{2}$

$\frac{1}{4}\|\mathbf{u}+\mathbf{v}\|^{2}-\frac{1}{4}\|\mathbf{u}-\mathbf{v}\|^{2}=\langle\mathbf{u}, \mathbf{v}\rangle$

Calculus 3

Chapter 6

Orthogonality and Least Square

Section 7

Inner Product Spaces

Vectors

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Hello there. So for this exercise, we need to show these expression here. So let's focus on the right hand side so we can regret this expression in a different way. So let's remember that the norm off any vector you is defined at the square root off the inner product off you with itself. Okay, so in this case, we're considering the squares off the norms. So that means we're just considering the inner product off the factor with itself. Okay, so this so, yeah, let us regret this expression here in terms off the inner products. Okay, so let's take each component page component. So first, we're going to just take into account this expression here, so this will be equal to the inner product off you plus v u plus V, divided four. But this is just you squared, plus two times you be Plus, So here is you with you on the here is a V to be divided four. Okay, on the other expression here. So this is the first expression. Then let's consider the second one. So in the second one is do minus V, divided four. So these he's just taking in their product off U minus V with U minus v. Divided four on This is equals to you with you minus two times the product off you with V on plus the in their product off B with be divided four. Okay, so we got these two expressions on. We need to subtract them. Okay, so at the end, we got the following we got you with you. Invited four plus one half off the inner product with off you with V plus the be divided four on from the other expression we got minus you, you divided four. Plus, I'm going to put this on the next line plus one half you movie and minus dinner product off. We with itself divided for So here is clear that this expression is going to canceled with this expression here on this one. With this one on, these two expressions here are going to sum up on this will become just you, V. That is exactly what we want to show. And that's it.

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