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The length of the perpendicular to the plane $\bar{r} \cdot(\hat{\imath}-2 \hat{\jmath}+3 \hat{k})=14$ from the origin is
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$\sqrt{14}$ units
Length of $\perp$ er from the point $A(\bar{a})$ to the plane $\bar{r} \cdot \bar{n}=p$ is $\frac{|\bar{a} \cdot \bar{n}-p|}{|\bar{n}|}$
Here $\overline{\mathrm{a}}=0 \hat{\mathrm{i}}+0 \hat{\mathrm{j}}+0 \hat{\mathrm{k}}$ and $\overline{\mathrm{n}}=\hat{\mathrm{i}}-2 \hat{\mathrm{j}}+3 \hat{\mathrm{k}}$
$\therefore \overline{\mathrm{a}} \cdot \overline{\mathrm{n}}=0 \quad$ and $|\overline{\mathrm{n}}|=\sqrt{1+4+9}=\sqrt{14}$
Hence required distance is $\frac{|0-14|}{\sqrt{14}}=\sqrt{14}$
Here $\overline{\mathrm{a}}=0 \hat{\mathrm{i}}+0 \hat{\mathrm{j}}+0 \hat{\mathrm{k}}$ and $\overline{\mathrm{n}}=\hat{\mathrm{i}}-2 \hat{\mathrm{j}}+3 \hat{\mathrm{k}}$
$\therefore \overline{\mathrm{a}} \cdot \overline{\mathrm{n}}=0 \quad$ and $|\overline{\mathrm{n}}|=\sqrt{1+4+9}=\sqrt{14}$
Hence required distance is $\frac{|0-14|}{\sqrt{14}}=\sqrt{14}$
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