If \(|\vec{a}\times \vec{b}|^2+|\vec{a}\cdot \vec{b}|^{2}=144\) and \(|\vec{a}|=4\), then \(|\vec{b}|\) is equal to

\(3\)

\(|\vec{a}\times \vec{b}|^2+|\vec{a}\cdot \vec{b}|^{2}≡|a|^2|b|^2(\sin^2θ+\cos^2θ)\)

$⇒|a|^2|b|^2=144$

\(\begin{aligned}|\vec{a}\times \vec{b}|^2+|\vec{a}\cdot \vec{b}|^2&=|a|^2|b|^2\\ 144&=16\cdot |\vec{b}|^{2}\\ |\vec{b}|^{2}&=9\\ |\vec{b}|&=3\end{aligned}\)