If \(|\vec{a}|\)=8,\(|\vec{b}|\)=3 and \(|\vec{a} ×\vec{ b}|\)=12, then the value \(\vec{a}.\vec{b}\) is
Answer & explanation
Correct answer: option 2
\(|\vec{a} ×\vec{b}|=12\)
$|\vec{a}||\vec{ b}|sinθ=12$
$sinθ=\frac{12}{|\vec{a}||\vec{ b}|}⇒sinθ=\frac{12}{8×3}=\frac{1}{2}$
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).