Question

Vectors $a \hat{i}+b \hat{j}+\hat{k}$ and $2 \hat{i}-3 \hat{j}+4 \hat{k}$ are perpendicular to each other when $3 a+2 b=7$, the ratio of $a$ to $b$ is $\frac{x}{2}$ The value of $x$ is __

Answer 1

For two perpendicular vectors
$ (a \hat{i}+b \hat{j}+\hat{k}) \cdot(2 \hat{i}-3 \hat{j}+4 \hat{k})=0 $
$2 a-3 b+4=0$
On solving, $2 a-3 b=-4$
Also given
$3 a +2 b =7$
We get $a =1, b =2$
$ \frac{a}{b}=\frac{x}{2} \Rightarrow x=\frac{2 a}{b}=\frac{2 \times 1}{2}$
$\Rightarrow x=1$