What is the condition for the vectors $2i+\text{3j}-\text{4k}\text{and}3i-\text{aj}+\text{bk}$ to be parallel

$\overrightarrow{B}=\left(3\hat{i}-\text{a}\hat{j}+\text{b}\hat{k}\right)$

When these two vectors are parallel, then angle between them is zero.

$\theta =0$

$\overrightarrow{A}\times \overrightarrow{B}=\left|\overrightarrow{A}\right|\left|\overrightarrow{B}\right|\sin \theta$

$\left(2\hat{i}+3\hat{j}-4\hat{k}\right)\times \left(3\hat{i}-\text{a}\hat{j}+\text{b}\hat{k}\right)=0$

$\left(3\text{b}-4\text{a}\right)\hat{i}-\left(2\text{b}+12\right)\hat{j}+\left(-2\text{a}-9\right)\hat{k}=0$

$3\text{b}=4\text{a}$

$\text{3}\left(-6\right)=4\text{a}$

$-\text{2b}-\text{12}=\text{0}$

$-\text{2b}=\text{12}$

$\text{b}=-\text{6}$

$3\left(-6\right)=4\text{a}$

$-\text{18}=\text{4a}$

$\frac{-18}{4}=\text{a}$

$\frac{-9}{2}=\text{a}$

$\text{b}=-\text{6}$