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

$2\hat{i} + 4\hat{j} + 4\hat{k}$

The correct answer is Option (4) → $2\hat{i} + 4\hat{j} + 4\hat{k}$ ##

Let ${a} = 2\hat{{i}} - \hat{{j}} + 2\hat{{k}}$ and ${b} = 4\hat{{i}} - \hat{{j}} + 3\hat{{k}}$

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

${a} \times {b} = \begin{vmatrix} \hat{{i}} & \hat{{j}} & \hat{{k}} \\ 2 & -1 & 2 \\ 4 & -1 & 3 \end{vmatrix}$

$= \hat{{i}}(-3 + 2) - \hat{{j}}(6 - 8) + \hat{{k}}(-2 + 4)$

$= -\hat{{i}} + 2\hat{{j}} + 2\hat{{k}} = {r} \quad [\text{say}]$

A vector of magnitude 6 in the direction of ${r}$

$= \frac{{r}}{|{r}|} \cdot 6 = \frac{-\hat{{i}} + 2\hat{{j}} + 2\hat{{k}}}{\sqrt{(-1)^2 + 2^2 + 2^2}} \cdot 6$

$= \frac{-6}{3}\hat{{i}} + \frac{12}{3}\hat{{j}} + \frac{12}{3}\hat{{k}}$

$= -2\hat{{i}} + 4\hat{{j}} + 4\hat{{k}}$