Q:

Dividing an equation by a vector

I am given the two equations
$$\bar{x} \times \bar{b} = \bar{a}$$
$$\bar{a} + \bar{x} = \lambda\bar{c}$$
I multiple the second by $\bar{b}$:
$$\bar{a} \times \bar{b} + \bar{x} \times \bar{b} = \lambda\bar{c} \times \bar{b}$$
and substitute the first one:
$$\bar{a} \times \bar{b} + \bar{a} = \lambda\bar{c} \times \bar{b}$$
divide to get $\lambda$ by itself:
$$\frac{\bar{a} \times \bar{b} + \bar{a}}{\bar{c} \times \bar{b}} = \lambda$$
and substitute back into the second equation and subtract $\bar{a}$:
$$\bar{x} = (\frac{\bar{a} \times \bar{b} + \bar{a}}{\bar{c} \times \bar{b}})\bar{c} - \bar{a}$$
However I'm told the answer is:
$$\bar{x} = (\frac{|\bar{a} \times \bar{b} + \bar{a}|}{|\bar{c} \times \bar{b}|})\bar{c} - \bar{a}$$
Where did the modulus come from? I'm guessing it's something to do with dividing by vectors?

A:

You see that
$$a\times b+a=\lambda c\times b$$
implies two vectors are collinear. Hence you can write
$$\lambda=\pm\frac{|a\times b+a|}{|c\times b|}$$

