$|V|=?$ if $2\vec{V}+(\vec{V}\times(\vec{i}+2\vec{j}))=2\vec{i}+\vec{k} $

Question

I am given $$2\vec{V}+(\vec{V}\times(\vec{i}+2\vec{j})=2\vec{i}+\vec{k} $$ and I need to find $$|V|=?$$ , I took dot product of equation with $\vec{V}$ but got stuck in $$2V^2 = \vec{V}.(2\vec{i}+\vec{k})$$ but there seems no way out similarly with cross product , what to do ?

Answer 1

There is probably a quicker way but if you write $\mathbf{V}=a\,\mathbf{i}+b\,\mathbf{j}+c\,\mathbf{k}$ and compute the left-hand-side and compare is to the right-hand side you will generate equations in $a,b$ and $c$ which can be solved.