Projection of u onto v and v onto u

Given the vector \(\displaystyle{u}={ < }−{2},{6},{4}{>}\) and a vector v such that the vector projection of u onto v is \(\displaystyle{ < }{2},{4},{4}{>}\), and the vector projection of v onto u is \(\displaystyle{ < }-{8},{24},{16}{>}\). What is the vector v?

Let \(\displaystyle\vec{{{v}}}={ < }{a},{b},{c}{>}\)

Projection of \(\displaystyle\vec{{{v}}}\ {o}{n}\to\ \vec{{{u}}}\) is given by:

\(\displaystyle{p}{r}{o}{j}_{{{u}}}{v}={\frac{{\vec{{{u}}}.\vec{{{v}}}}}{{{\left|{u}\right|}^{{{2}}}}}}\vec{{{u}}}={\frac{{\vec{{{u}}}.\vec{{{v}}}}}{{{\left(-{2}\right)}^{{{2}}}+{6}^{{{2}}}+{4}^{{{2}}}}}}{ < }-{2},{6},{4}{>}\)

\(\displaystyle{ < }-{8},{24},{16}\ge{\frac{{\vec{{{u}}}.\vec{{{v}}}}}{{{\left(-{2}\right)}^{{{2}}}+{6}^{{{2}}}+{4}^{{{2}}}}}}{ < }-{2},{6},{4}{>}\)

\(\displaystyle\Rightarrow{4}{ < }-{2},{6},{4}\ge{\frac{{\vec{{{u}}}.\vec{{{v}}}}}{{{4}+{36}+{16}}}}{ < }-{2},{6},{4}{>}\)

\(\displaystyle\Rightarrow{ < }-{2},{6},{4}\ge{\frac{{\vec{{{u}}}.\vec{{{v}}}}}{{{224}}}}{ < }-{2},{6},{4}{>}\)

On comparing \(\displaystyle{\frac{{\vec{{{u}}}.\vec{{{v}}}}}{{{224}}}}={1}\)

\(\displaystyle\Rightarrow\vec{{{u}}}.\vec{{{v}}}={224}\)

\(\displaystyle{p}{r}{o}{j}_{{{v}}}{u}={\frac{{\vec{{{u}}}.\vec{{{v}}}}}{{{\left|{v}\right|}^{{{2}}}}}}\vec{{{v}}}={\frac{{{224}}}{{{\left|{v}\right|}^{{{2}}}}}}{ < }{a},{b},{c}{>}\)

\(\displaystyle\Rightarrow{ < }{2},{4},{4}\ge{\frac{{{224}}}{{{\left|{v}\right|}^{{{2}}}}}}{ < }{a},{b},{c}{>}\)

Dividing both sides by 2 we get:

\(\displaystyle\Rightarrow{ < }{1},{2},{2}\ge{\frac{{{112}}}{{{\left|{v}\right|}^{{{2}}}}}}{ < }{a},{b},{c}{>}\)