question_answer

The distance from the point \[-\mathbf{i}+2\mathbf{j}+6\mathbf{k}\] to the straight line through the point (2, 3, ?4) and parallel to the vector \[6\mathbf{i}+3\mathbf{j}-4\mathbf{k}\] is

A)            7

B)            10

C)            9

D)            None of these

Correct Answer: A

Solution :

           We have \[\overrightarrow{AP}=-3\mathbf{i}-\mathbf{j}+10\mathbf{k}\]                    \  \[|\overrightarrow{AP}|=\sqrt{9+1+100}=\sqrt{110}\]                                  \[AN=\]Projection of \[\overrightarrow{AP}\] on \[6\mathbf{i}+3\mathbf{j}-4\mathbf{k}\]            \[=\left| \frac{\overrightarrow{AP}.(6\mathbf{i}+3\mathbf{j}-4\mathbf{k})}{|6\mathbf{i}+3\mathbf{j}-4\mathbf{k}|} \right|\]\[=\left| \frac{-18-3-40}{\sqrt{61}} \right|=\sqrt{61}\]                    \ \[PN=\sqrt{A{{P}^{2}}-A{{N}^{2}}}\]\[=\sqrt{110-61}=7\].