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

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

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Answer 1 · 2016-05-31T19:29:30

See demonstration below, please.

Explanation:

Let ${\to v}_{1} = (a, b, c)$ $vec v_1=(a,b,c)$ and ${\to v}_{2} = (x, y, z)$ $vec v_2 = (x,y,z)$
we know that

${\to v}_{1} . {\to v}_{2} = ∥ ∥ {\to v}_{1} ∥ ∥ ∥ ∥ {\to v}_{2} ∥ ∥ cos (ˆ {\to v}_{1}, {\to v}_{2})$ $vec v_1.vec v_2 = norm(vec v_1) norm(vec v_2) cos(hat(vec v_1,vec v_2))$

and

$cos (ˆ {\to v}_{1}, {\to v}_{2}) = \frac{a x + b y + c z}{\sqrt{a^{2} + b^{2} + c^{2}} \sqrt{x^{2} + y^{2} + z^{2}}} = \frac{20}{4 \times 5} = 1$ $cos(hat(vec v_1,vec v_2)) = (a x+b y +c z)/(sqrt(a^2+b^2+c^2)sqrt(x^2+y^2+z^2)) = 20/(4 times 5)=1$

So ${\to v}_{1}$ $vec v_1$ and ${\to v}_{2}$ $vec v_2$ are aligned or ${\to v}_{1} = λ {\to v}_{2}$ $vec v_1 = lambda vec v_2$

and $λ = \frac{4}{5}$ $lambda = 4/5$

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

1 Answer

Explanation:

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Given a^2+b^2+c^2=16;x^2+y^2+z^2=25 and ax+by+cz=20a2+b2+c2=16;x2+y2+z2=25andax+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/zax=by=cza/x=b/y=c/z ? Find also the value of each ratio.

1 Answer

Explanation:

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