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.

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Answer 1 · 2016-05-31T19:29:30

See demonstration below, please.

Let $vec v_1=(a,b,c)$ and $vec v_2 = (x,y,z)$
we know that

$vec v_1.vec v_2 = norm(vec v_1) norm(vec v_2) cos(hat(vec v_1,vec v_2))$

and

$cos(hat(vec v_1,vec v_2)) = (a x+b y +c z)/(sqrt(a^2+b^2+c^2)sqrt(x^2+y^2+z^2)) = 20/(4 times 5)=1$

So $vec v_1$ and $vec v_2$ are aligned or $vec v_1 = lambda vec v_2$

and $lambda = 4/5$

Given a^2+b^2+c^2=16;x^2+y^2+z^2=25 and ax+by+cz=20a^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/za/x=b/y=c/z ? Find also the value of each ratio.