What is the condition for the vectors $2\hat{\mathrm{i}}+3\hat{\mathrm{j}}-4\mathrm{k} \mathrm{and} 3\hat{\mathrm{i}}-\mathrm{a}\hat{\mathrm{j}}+\mathrm{b}\hat{\mathrm{k}}$ to be parallel ?

If the cross product of two vectors is zero, then the two vectors must be parallel. i.e

$\overrightarrow{\mathrm{A}}\times \overrightarrow{\mathrm{B}}=0$

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

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

Given that the vector on the right hand is 0

$-12-2\mathrm{b}=04 \mathrm{and} -2\mathrm{a}-9=0$

$\Rightarrow \mathrm{b}=-6 \mathrm{and} \mathrm{a}=\frac{-9}{2}$

Hence the correct answer is $\mathrm{a}=\frac{-9}{2},\mathrm{b}=-6.$