If \(|\vec{a}|\)=8,\(|\vec{b}|\)=3 and \(|\vec{a} ×\vec{ b}|\)=12, then the value \(\vec{a}.\vec{b}\) is

12\(\sqrt { 3}\)

\(|\vec{a} ×\vec{b}|=12\)

$|\vec{a}||\vec{ b}|sinθ=12$

$sinθ=\frac{12}{|\vec{a}||\vec{ b}|}⇒sinθ=\frac{12}{8×3}=\frac{1}{2}$

θ = 30° or 150°

\(\vec{a}.\vec{b}=|\vec{a}||\vec{ b}|cosθ\)

$\vec{a}.\vec{b}=8×3×cos30°⇒8×3\frac{\sqrt{3}}{2}=12\sqrt{3}$ .....(i)

Similarly

\(\vec{a}.\vec{b}=|\vec{a}||\vec{ b}|cosθ\)

\(\vec{a}.\vec{b}=8×3×cos150°\)

\(\vec{a}.\vec{b}=-8×3×\frac{\sqrt{3}}{2}\)

\(\vec{a}.\vec{b}=-12\sqrt{3}\)

Option 2 is correct from eq. (i).