Hi, I need a hint to start this problem:
If $\displaystyle (a^2 +b^2+c^2)(x^2+y^2+z^2) = (ax+by+cz)^2$
prove that $\displaystyle \frac{x}{a} = \frac{y}{b} = \frac{z}{c} $
Thanks.
That's not a ratio... but since you asked for a hint...
Expand all the brackets and collect like terms, you should get...
$\displaystyle a^2y^2 + a^2z^2 + b^2x^2 + b^2z^2 + c^2x^2 + c^2y^2 = 2abxy + 2acxz + 2bcyz$
$\displaystyle b^2x^2 - 2abxy + a^2y^2 + a^2z^2 -2acxz + c^2x^2 + b^2z^2 - 2bcyz + c^2y^2 = 0$
Recall that $\displaystyle (p-q)^2 = p^2 - 2pq + q^2$? Using this we get...
$\displaystyle (bx - ay)^2 + (az - cx)^2 + (bz - cy)^2 = 0$
Now, assuming a, b, c, x, y and z are all real... we have 3 square terms adding to become 0. Since a square term is never non-negative... all the terms must be 0.
So we have $\displaystyle (bx - ay)^2 = 0$ which gives $\displaystyle bx = ay$ or $\displaystyle \frac{x}{a} = \frac{y}{b}$.
Similarly, $\displaystyle (az - cx)^2 = 0$ gives $\displaystyle \frac{z}{c}=\frac{x}{a}$,
and $\displaystyle (bz - cy)^2 = 0$ gives $\displaystyle \frac{z}{c} = \frac{y}{b}$.
Thus
$\displaystyle \frac{x}{a} = \frac{y}{b} = \frac{z}{c}$.