Question

Find the length of the perpendicular drawn from the point (5, 4, −1) to the line \[\overrightarrow{r} = \hat{i}  + \lambda\left( 2 \hat{i} + 9 \hat{j} + 5 \hat{k} \right) .\]

Answer 1

Let the point (5, 4, -1)   be P and the the point through which the line passes be Q (1, 0, 0).
The line is parallel to the vector  \[\overrightarrow{b} = 2 \hat{i} + 9 \hat{j} + 5 \hat{k} \] 

Now,  \[\overrightarrow{PQ} = - 4 \hat{i} - 4 \hat{j} + \hat{k} \] 

\[\therefore \overrightarrow{b} \times \overrightarrow{PQ} = \begin{vmatrix}\hat{i} & \hat{j}  & \hat{k} \\ 2 & 9 & 5 \\ - 4 & - 4 & 1\end{vmatrix}\]

\[ = 29 \hat{i} - 22 \hat{j} + 28 \hat{k} \]

\[ \Rightarrow \left| \overrightarrow{b} \times \overrightarrow{PQ} \right| = \sqrt{{29}^2 + \left( - 22 \right)^2 + {28}^2}\]

\[ = \sqrt{841 + 484 + 784}\]

\[ = \sqrt{2109}\]

\[\left| \overrightarrow{b} \right| = \sqrt{2^2 + 9^2 + 5^2}\]

\[ = \sqrt{4 + 81 + 25}\]

\[ = \sqrt{110}\]

\[d = \frac{\left| \overrightarrow{b} \times \overrightarrow{PQ} \right|}{\left| \overrightarrow{b} \right|}\]

\[ = \frac{\sqrt{2109}}{\sqrt{110}}\]

\[ = \sqrt{\frac{2109}{110}}\]