How to get ratio of a,b,c from 2 equations in a,b,c

Question

I have 2 equation in terms of $a,b$ and $c$ .

$3a + 10b + 5c =0$ and $4a + 6b + 2c =0$

I need to find a:b:c and answer is $\dfrac a5 = \dfrac {b}{-7} = \dfrac {c}{11}$

I want to know how to get that?

My attempt:

Given equations can be written in form

$$\left ( \begin{matrix} 3 & 10 & 5 \\ 4 & 6 & 2 \\ \end{matrix} \right ) \left ( \begin{matrix} a\\ b\\ c\\ \end{matrix} \right ) = \left ( \begin{matrix} 0 \\ 0\\ \end{matrix} \right ) $$

But here I don't have any idea how to proceed. I can do for 3×3 matrix and I thought it can be done in same way.

Thanks.

Answer 1

HINT:

$$3a+10b=-5c$$

$$4a+6b=-2c$$

Solve the two simultaneous equations for $a,b$ in terms of $c$

See this.

Answer 2

$(a\,b\,c)$ is orthogonal to both $(3\,10\,5)$ and $(4\,6\,2)$. The obvious choice for a common orthogonal vector in 3-space is the cross product.

Answer 3

Hint...first eliminate one of the letters, say $c$ so that you have one equation in $a$ and $ b$. Then sepatate these letters to either side of the equation and set each side equal to a parameter $\lambda$. You can then get each of $a,b,c$ in terms of $\lambda$ to get the set of ratios.

Note that this problem is equivalent to that of finding the line of intersection of two planes in three dimensional space.