Question

Find a vector of magnitude $19$ which is perpendicular to both the vectors $-\hat{j}+\hat{k}$ and $4\hat{i}-\hat{j}+8\hat{k}.$

Answer 1

Let the required vector be $a\hat{i}+b\hat{j}+c\hat{k}$

$\therefore (a\times 0)\hat{i}.\hat{i}-b\hat{j}.\hat{j}+c\hat{k}.\hat{k}=0$

$\Rightarrow –b+c=0$

$\Rightarrow b=c$

And

$4a-b+8c=0$

$\Rightarrow 4a+7b=0$

$\Rightarrow a=\frac{-7b}{4}$

$\Rightarrow \frac{a}{\frac{-7}{4}}=\frac{b}{1}=\frac{c}{1}$

$\therefore$ vector is $19\times \left(\frac{-\frac{7}{4}\hat{i}+\hat{j}+\hat{k}}{\sqrt{\frac{49}{16}+1+1}}\right)$

$=19\times \frac{4}{9}(-\frac{7}{4}\hat{i}+\hat{j}+\hat{k})$

$=19(-\frac{7}{9}\hat{i}+\frac{4}{9}\hat{j}+\frac{4}{9}\hat{k})$