Question

Find a vector of magnitude 6 , which is perpendicular to both the vectors $2\hat{i}-\hat{j}+2\hat{k}\text{and}4\hat{i}-\hat{j}+3\hat{k}.$

Answer 1

Let $2\hat{i}-\hat{j}+2\hat{k}\text{and}4\hat{i}-\hat{j}+3\hat{k}.$

So, any vector perpendicular to both the vectors $\overrightarrow{a}and\overrightarrow{b}$ is given by

$\overrightarrow{a}\times \overrightarrow{b}=\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ 2& -1& 2\\ 4& -1& 3\end{array}\right|=\hat{i}(-3+2)-\hat{j}(6-8)+\hat{k}(-2+4)=-\hat{i}+2\hat{j}+2\hat{k}=\overrightarrow{r}\left[say\right]$

A vector of magnitudes 6 in the direction of $\overrightarrow{r}$

$=\frac{\overrightarrow{r}}{|\overrightarrow{r}|}.6=\frac{-\hat{i}+2\hat{j}+2\hat{k}}{\sqrt{1^2+2^2+2^2}}.6=\frac{-3}{3}\hat{i}+\frac{12}{3}\hat{j}+\frac{12}{3}\hat{k}=-2\hat{i}+4\hat{j}+4\hat{k}$