If $\vec{a}=\hat{i}+2 \hat{j}+2 \hat{k},|\vec{b}|=5$ and the angle between $\vec{a}$ and $\vec{b}$ is $\frac{\pi}{6}$, then the area of the triangle formed by these two vectors as two sides is:

$\frac{15}{4}$ sq. unit

The correct answer is Option (3) → $\frac{15}{4}$ sq. unit

area of triangle = $\frac{|\vec a×\vec b|}{2}=\frac{|\vec a||\vec b|}{2}\sin\frac{\pi}{6}$

$=\frac{(\sqrt{1^2+2^2+2^2})×5}{2}\sin\frac{\pi}{6}$

$=\frac{3×5}{2×2}=\frac{15}{4}$ sq. unit