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 vectors inclined at an angle θ, then identify the correct option out of…

If $\vec{a} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k}$a=a1​i^+a2​j^​+a3​k^ and $\vec{b} = b_1\hat{i} + b_2\hat{j} + b_3\hat{k}$b=b1​i^+b2​j^​+b3​k^ are two non zero vectors inclined at an angle $\theta$θ, then identify the correct option out of the given options.
(a) $\cos\theta = \frac{\vec{a} \cdot \vec{b}}{|\vec{a}| \cdot |\vec{b}|}$cosθ=∣a∣⋅∣b∣a⋅b​
(b) $\vec{a}$a and $\vec{b}$b are perpendicular, if $a_1 b_1 + a_2 b_2 + a_3 b_3 = 0$a1​b1​+a2​b2​+a3​b3​=0
(c) $\vec{a}$a and $\vec{b}$b are perpendicular, if $\frac{a_1}{b_1} = \frac{a_2}{b_2} \neq \frac{c_1}{c_2}$b1​a1​​=b2​a2​​=c2​c1​​
(d) for $\theta = \pi$θ=π, $\vec{a} \times \vec{b} = 0$a×b=0
(e) $\cos\theta = \frac{|\vec{a} \times \vec{b}|}{|\vec{a}| \cdot |\vec{b}|}$cosθ=∣a∣⋅∣b∣∣a×b∣​
Choose the most appropriate answer from the options given below