Giải thích các bước giải:
\[\begin{array}{l}
\left( {{x^2} + {y^2} + {z^2}} \right)\left( {{a^2} + {b^2} + {c^2}} \right) = {\left( {ax + by + cz} \right)^2}\\
\Leftrightarrow {x^2}{a^2} + {x^2}{b^2} + {x^2}{c^2} + {y^2}{a^2} + {y^2}{b^2} + {y^2}{c^2} + {z^2}{a^2} + {z^2}{b^2} + {z^2}{c^2} = {a^2}{x^2} + {b^2}{y^2} + {c^2}{z^2} + 2\left( {ax.by + by.cz + cz.ax} \right)\\
\Leftrightarrow {x^2}{b^2} + {x^2}{c^2} + {y^2}{a^2} + {y^2}{c^2} + {z^2}{a^2} + {z^2}{b^2} - 2\left( {ax.by + by.cz + cz.ax} \right) = 0\\
\Leftrightarrow \left( {{x^2}{b^2} - 2abxy + {y^2}{a^2}} \right) + \left( {{x^2}{c^2} - 2caxz + {a^2}{z^2}} \right) + \left( {{y^2}{c^2} - 2bycz + {b^2}{z^2}} \right) = 0\\
\Leftrightarrow {\left( {xb - ya} \right)^2} + {\left( {xc - az} \right)^2} + {\left( {yc - bz} \right)^2} = 0\\
\Leftrightarrow \left\{ \begin{array}{l}
xb - ya = 0\\
xc - az = 0\\
yc - bz = 0
\end{array} \right. \Leftrightarrow \left\{ \begin{array}{l}
xb = ya\\
cx = az\\
yc = bz
\end{array} \right. \Leftrightarrow \left\{ \begin{array}{l}
\frac{x}{a} = \frac{y}{b}\\
\frac{x}{a} = \frac{z}{c}\\
\frac{y}{b} = \frac{z}{c}
\end{array} \right. \Rightarrow \frac{x}{a} = \frac{y}{b} = \frac{z}{c}
\end{array}\]
