Question

Let a $\overrightarrow{a}=\hat{1}+4\hat{J}+2\hat{k},\overrightarrow{b}=3\hat{i}-2\hat{j}+7\hat{k}andc=2\hat{i}-\hat{j}+4\hat{k}.$ find a vector $\overrightarrow{d}$ which is perpendicular to both $\overrightarrow{a}$ and $\overrightarrow{b}$ and $\overrightarrow{c}.\overrightarrow{d}=27.$

Answer 1

Since $\overrightarrow{d}$ is perpendicular to both $\overrightarrow{a}$ and $\overrightarrow{b}$ $\therefore \overrightarrow{d}$ is parallel to $\overrightarrow{a}\times \overrightarrow{b}$.

So $\overrightarrow{d}=\lambda (\overrightarrow{a}\times \overrightarrow{b})=\lambda \left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ 1& 4& 2\\ 3& -2& 7\end{array}\right|=32\lambda \hat{i}-\lambda \hat{j}-14\lambda \hat{k}$

Also $\hat{c}.\hat{d}=(2\hat{i}-\hat{j}+4\hat{k}).(32\lambda \hat{i}-\lambda \hat{j}-14\lambda \hat{k})=27\therefore \lambda =3$

Therefore, $\overrightarrow{r}=96\hat{i}-3\hat{j}-42\hat{k}.$