Eigenvectors and the dot product

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Homework Statement

Suppose the the matrix A is symmetric, meaning that
A =

a b
b d

Show that for any symmetric matrix A there are always real eigenvalues. Also, show that
the eigenvectors corresponding to two dierent eigenvalues are always orthogonal; that is,
if V1 and V2 are the eigenvectors for eigenvalues \lambda1 and \lambda2, with \lambda1 not equal to \lambda2, then the dot
product V1 *
V2 = 0.
HINT: Compare \lambda1V1*V2 to V1*\lambda2V2 .

Homework Equations

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The Attempt at a Solution

I was able to show that any eigenvalue from this matrix would always be positive. That wasn't too bad. Its the second part that is tripping me up. I don't get how the hint is really all that helpful. I want to find values of V1 and V2 don't I? I'm confused.

 

[vela]

You don't need to find the eigenvectors explicitly. All you need to show is that their dot product is zero.

Consider <Av₁,v₂> and <v₁,Av₂> and, using the fact that A is symmetric, show that they are equal.