The distance of the point having position vector

Question:

The distance of the point having position vector $-\hat{\mathrm{i}}+2 \hat{\mathrm{j}}+6 \hat{\mathrm{k}}$ from the straight line passing through the point $(2,3,-4)$ and parallel to the vector, $6 \hat{i}+3 \hat{j}-4 \hat{k}$ is :

  1. 7

  2. $4 \sqrt{3}$

  3. $2 \sqrt{13}$

  4. 6


Correct Option: 1

Solution:

$\mathrm{AD}=\left|\frac{\overrightarrow{\mathrm{AP}} \cdot \overrightarrow{\mathrm{n}}}{|\overrightarrow{\mathrm{n}}|}\right|=\sqrt{61}$

$\Rightarrow \quad P D=\sqrt{A P^{2}-A D^{2}}=\sqrt{110-61}=7$

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