Question

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.

Answer 1

See demonstration below, please.

Explanation:

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`