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.

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.

Eva Abramson · Accepted Answer

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#

Kennedy Adams · Answer

undefined To prove $ \frac{a}{x} = \frac{b}{y} = \frac{c}{z} $, we can use the given equation $ ax + by + cz = 20 $. Rearrange it to express $ a $, $ b $, and $ c $ in terms of $ x $, $ y $, and $ z $, then substitute these expressions into $ a^2 + b^2 + c^2 = 16 $ and $ x^2 + y^2 + z^2 = 25 $. After simplifying, we get $ \frac{a}{x} = \frac{b}{y} = \frac{c}{z} = \frac{4}{5} $.

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.