# The magnitude of the projection of the vector $2\hat{i}+3\hat{j}+\hat{k}$ on the vector perpendicular to the plane containing the vectors $\hat{i}+\hat{j}+\hat{k}$ and $\hat{i}+2\hat{j}+3\hat{k}$ is :   Option 1) $\sqrt{\frac{3}{2}}$ Option 2) $\sqrt{6}$ Option 3) $\frac{\sqrt{3}}{2}$   Option 4) $3\sqrt{6}$

Normal vector to plane containing vector $\hat{i}+\hat{j}+\hat{k}$

and $\hat{i}+2\hat{j}+3\hat{k}$ is

$\hat{n}=(\hat{i}+\hat{j}+\hat{k})\times(\hat{i}+2\hat{j}+3\hat{k})$

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

Projection of $2\hat{i}+3\hat{j}+\hat{k}$ on $\hat{n}$

$=\left | \frac{(2\hat{i}+3\hat{j}+\hat{k})(\hat{i}-2\hat{j}+\hat{k})}{\sqrt{1+4+1}} \right |$

$=\frac{3}{\sqrt{6}}=\sqrt{\frac{3}{2}}$

Option 1)

$\sqrt{\frac{3}{2}}$

Option 2)

$\sqrt{6}$

Option 3)

$\frac{\sqrt{3}}{2}$

Option 4)

$3\sqrt{6}$

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