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

$\begin{array}{1 1} (A)\;2(-\hat i+\hat j+3\hat k) \\(B)\;-\hat i-2\hat j+2\hat k \\ (C)\;-2\hat i+4\hat j+4\hat k \\ (D)\;2(-\hat i-\hat j-3\hat k) \end{array}$

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• A unit vector perpendicular to two vectors is given by $\hat n =\large \frac{\overrightarrow a \times \overrightarrow b}{|\overrightarrow a \times \overrightarrow b|}$
Let $\overrightarrow a= 2\hat i - \hat j + 2\hat k$ and $\overrightarrow b=4\hat i -\hat j + 3\hat k$
Let us first determine $\overrightarrow a\times\overrightarrow b$
$\overrightarrow a \times \overrightarrow b=\begin{vmatrix} \hat i & \hat j & \hat k \\ 2 & -1 & 2 \\ 4 & -1 & 3 \end{vmatrix}$
On expanding we get,
$=\hat i(-3+2)-\hat j(6-8)+\hat k(-2+4)$
$= -\hat i + 2\hat j + 2\hat k$
$|\overrightarrow a\times\overrightarrow b|=\sqrt{(-1^2+2^2+2^2)}=\sqrt {1+4+4}=3$
Hence the unit vector perpendicular to the given vectors of magnitude $6$ is $3$
$= \large\frac{6}{3}$$\bigg( -\hat i + 2\hat j + 6\hat k \bigg)$
$=2(-\hat i+\hat j+3\hat k)$