If the vectors $2\widehat{i}+3\widehat{j}-4\widehat{k}$ and $3\widehat{i}-a\widehat{j}+b\widehat{k}$ are parallel, find the values of  $a$ and $b$. 

Two vectors are parallel if the cross product of them is zero. ie $\overrightarrow{A}\times \overrightarrow{B}=0$

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

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

$-2b-12=0\Rightarrow b=-6$

$3b-4a=0$

$3b=4a$.

$3(-6)=4a$

$a=-9/2,b=-6$

Alternatively: 
Two vectors $\overrightarrow{A}\text{\hspace{0.17em}\hspace{0.17em}and\hspace{0.17em}\hspace{0.17em}}\overrightarrow{B}$  are parallel if

$\frac{A_x}{B_x}=\frac{A_y}{B_y}=\frac{A_z}{B_z}$

$\Rightarrow \frac{2}{3}=\frac{3}{-a}=\frac{-4}{b}\Rightarrow a=\frac{-9}{2},b=-6$