# Given a^2+b^2+c^2=16;x^2+y^2+z^2=25 and ax+by+cz=20 for a,b,c being real. How will you prove a/x=b/y=c/z ? Find also the value of each ratio.

May 31, 2016

#### Explanation:

Let ${\vec{v}}_{1} = \left(a , b , c\right)$ and ${\vec{v}}_{2} = \left(x , y , z\right)$
we know that

${\vec{v}}_{1.} {\vec{v}}_{2} = \left\lVert {\vec{v}}_{1} \right\rVert \left\lVert {\vec{v}}_{2} \right\rVert \cos \left(\hat{{\vec{v}}_{1} , {\vec{v}}_{2}}\right)$

and

$\cos \left(\hat{{\vec{v}}_{1} , {\vec{v}}_{2}}\right) = \frac{a x + b y + c z}{\sqrt{{a}^{2} + {b}^{2} + {c}^{2}} \sqrt{{x}^{2} + {y}^{2} + {z}^{2}}} = \frac{20}{4 \times 5} = 1$

So ${\vec{v}}_{1}$ and ${\vec{v}}_{2}$ are aligned or ${\vec{v}}_{1} = \lambda {\vec{v}}_{2}$

and $\lambda = \frac{4}{5}$