Cauchy - Schwartz inequality states that: If u,v are vectors such that $u = (u_{1}, u_{2}, u_{3}, .........u_{n})$ and $v = (v_{1}, v_{2}, v_{3},........v_{n})$ in $R^n$ in a real inner product space then

$$|u * v| \leq ||u|| * ||v||$$

Given u = (-4, 2, 1) & v = (8, -4, 2)

$$\therefore ||u|| = \sqrt{(-4)^2 + (2)^2 + (1)^2} = \sqrt{16 + 4 + 1} = \sqrt{21} \\ ||v|| = \sqrt{(8)^2 + (-4)^2 + (-2)^2} = \sqrt{64 + 16 + 4} = \sqrt{84} \\ \therefore ||u|| * ||v|| = \sqrt{21} * \sqrt{84} = \sqrt{21} * \sqrt{21 * 4} = 21 * 2 = 42 $$

Now, $$|u * v| = |u_{1}v_{1} + u_{2}v_{2} + u_{3}v_{3} + u_{4}v_{4}| = |(-4)(8) + (2)(-4) + (1)(-2)| \\ = |-32 - 8 - 2| = 42 \\ \therefore ||u|| * ||v|| = |u * v|$$

Hence, Cauchy- Schwartz inequality holds good for the vectors u & v