If a=a_1 i+a_2 j+a_3 k and b=b_1 i+b_2 j+b_3 k are two non zero vecto…

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