# Homework Help: Inner Product Question

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1. Jul 28, 2017

### Onezimo Cardoso

1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

I could show that the first of the three properties are valid for any value of a,b,c but I couldn’t find a way to show the forth one.

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2. Jul 28, 2017

### Ray Vickson

Hint: if ${\bf x} = (x_1,0)$, it is easy to show that $\langle {\bf x,x} \rangle > 0$ for any $x_1 \neq 0$. Also: for $x_2 \neq 0$ we have $(x_1,x_2) = x_2(x_1/x_2,1)$, so $\langle {\bf x,x} \rangle = x_2^2 \langle {\bf u,u} \rangle$, where ${\bf u} = (x_1/x_2,1) \equiv (t,1)$.

3. Jul 29, 2017

### ehild

4. Jul 29, 2017

### StoneTemplePython

are you allowed to use spectral theory here?

$\begin{bmatrix} a & b\\ b & c \end{bmatrix}$

is real symmetric. If the determinant is positive (one of your conditions in the iff) and the trace is positive (implied by $a \gt 0$... why?) then it this tells you...

5. Jul 31, 2017

### Onezimo Cardoso

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6. Jul 31, 2017

### ehild

If you factorize Δ you get Δ=4x22(b2-4ac)

7. Aug 1, 2017

### Ray Vickson

Why do you suppose I wrote $\langle {\bf x,x} \rangle = x_2^2 \langle {\bf u,u} \rangle$, where ${\bf u} = (t,1)?$ The fact that $t = x_1/x_2$ does not really matter at all if all you want to know is the sign of $\langle {\bf u,u} \rangle$---think about it!