Maximize $ax + by + cz$ given $x^2 + y^2 + z^2 = k^2$. Write the answer as the Schwartz inequality for dot products $(a, b, c) \cdot (x, y, z) \le \_\_\_\_\_\_\_\_ \ k$.
I'm stuck on this problem. I get that $ax + by + cz$ is the same as $(a, b, c) \cdot (x, y, z)$, but I'm not sure how to approach this problem.
Schwartz inequality says that $$(a,b,c)\cdot(x,y,z)\leq \|(a,b,c) \|\ \|(x,y,z)\|. $$ In this case $\|(a,b,c)\|=\sqrt{a^2+b^2+c^2}$, so it becomes $$ (a,b,c)\cdot(x,y,z)\leq \sqrt{a^2+b^2+c^2}\sqrt{x^2+y^2+z^2}=\left(\sqrt{a^2+b^2+c^2}\right)k.$$
Moreover schwartz inequality becomes an equality if and only if there exist $\lambda$ such that $(a,b,c)=\lambda(x,y,z)$.