Example 1 Suppose $a^{2}+b^{2}+c^{2}=4, x^{2}+y^{2}+z^{2}=9$, then the range of values for $a x+b y+c z$ is $\qquad$ (Example 9 from [1])
By Cauchy-Schwarz inequality, we have $4 \times 9=\left(a^{2}+b^{2}+c^{2}\right)\left(x^{2}\right.$ $\left.+y^{2}+z^{2}\right) \geqslant(a x+b y+c z)^{2}, \therefore-6 \leqslant a x+b y+c z$ $\leqslant 6$