If $ax+by+cz=k$,prove that the minimum value of $x^2+y^2+z^2$ is $\frac{k^2}{a^2+b^2+c^2}$ If $ax+by+cz=k$,prove that the minimum value of $x^2+y^2+z^2$ is $\frac{k^2}{a^2+b^2+c^2}$.
I know that this problem can be solved by Cauchy Schwartz.How can i find its minimum value using multivariable calculus or by other methods.Please help me.

$$(x^2+y^2+z^2)(a^2+b^2+c^2)-(ax+by+cz)^2 = (ay-bx)^2+(bz-cy)^2+(cx-az)^2 \ge 0$$
so $$x^2+y^2+z^2\ge\frac{(ax+by+cz)^2}{a^2+b^2+c^2}=\frac{k^2}{a^2+b^2+c^2}$$
This is in fact an elemental proof of Cauchy–Schwarz_inequality on $\mathbb{R}^3$ 
