If $\vec{a}=a_1 \hat{i}+a_2 \hat{j}+a_3 \hat{k}$ and $\vec{b}=b_1 \hat{i}+b_2 \hat{j}+b_3 \hat{k}$ are two non zero vectors inclined at an angle $\theta$, then identify the correct option out of the given options.

(a) $\cos \theta=\frac{\vec{a} \cdot \vec{b}}{|\vec{a}| \cdot|\vec{b}|}$
(b) $\vec{a}$ and $\vec{b}$ are perpendicular, if $a_1 b_1+a_2 b_2+$ $a_3 b_3=0$
(c) $\vec{a}$ and $\vec{b}$ are perpendicular, if $\frac{a_1}{b_1}=\frac{a_2}{b_2} \neq \frac{c_1}{c_2}$
(d) for $\theta=\pi, \vec{a} \times \vec{b}=0$
(e) $\cos \theta=\frac{|\vec{a} \times \vec{b}|}{|\vec{a}| \cdot|\vec{b}|}$

Choose the most appropriate answer from the options given below

(a), (b) and (d) only

(a) → correct

(b) → correct

(c) → incorrect for perpendicularity $a_1b_1+a_2b_2+a_3b_3=0$

(d) → $\theta=\pi$

$\vec{a} \times \vec{b}=|\vec{a}| . |\vec{b}| \sin \pi \hat{n}$

$=0$ → correct

(e) → incorrect

as  $\sin \theta = \frac{|\vec{a} × \vec{b}|}{|\vec{a}|. |\vec{b}|}$

Option: 1