Question

Find the foot of the perpendicular drawn from the point  \[\hat{i} + 6 \hat{j} + 3 \hat{k} \]  to the line  \[\overrightarrow{r} = \hat{j} + 2 \hat{k} + \lambda\left( \hat{i} + 2 \hat{j} + 3 \hat{k}  \right) .\]  Also, find the length of the perpendicular

Answer 1

Let L be the foot of the perpendicular drawn from the point P ( \[\hat{i} + 6 \hat{j} + 3 \hat{k}\] ) to the line \[\overrightarrow{r} = \hat{j} + 2 \hat{k} + \lambda\left( \hat{i} + 2 \hat{j} + 3 \hat{k}  \right) .\] 

Let the position vector L be  \[\overrightarrow{r} = \hat{j} + 2 \hat{k} + \lambda\left( \hat{i}  + 2 \hat{j}  + 3 \hat{k} \right) = \lambda \hat{i} + \left( 1 + 2\lambda \right) \hat{j} + \left( 2 + 3\lambda \right) \hat{k} \] 

Now, 

\[\overrightarrow{PL} = \text{ Position vector of L - Position vector of P } \]

\[ \Rightarrow \overrightarrow{PL} = \left\{ \lambda \hat{i} + \left( 1 + 2\lambda \right) \hat{j} + \left( 2 + 3\lambda \right) \hat{k} \right\} - \left( \hat{i}  + 6 \hat{j}  + 3 \hat{k}  \right)\]

\[ \Rightarrow \overrightarrow{PL} = \left( \lambda - 1 \right) \hat{i} + \left( 2\lambda - 5 \right) \hat{j} + \left( 3\lambda - 1 \right) \hat{k} . . . (2)\]

Since

\[\overrightarrow{PL}\]  is perpendicular to the given line, which is parallel to  \[\overrightarrow{b} = \hat{i} + 2 \hat{j} + 3 \hat{k} \]   

we have  , 

\[\overrightarrow{PL} . \overrightarrow{b} = 0\]

\[ \Rightarrow \left\{ \left( \lambda - 1 \right) \hat{i} + \left( 2\lambda - 5 \right) \hat{j} + \left( 3\lambda - 1 \right) \hat{k} \right\} . \left( \hat{i}  + 2 \hat{j} + 3 \hat{k}  \right) = 0 \]

\[ \Rightarrow 1\left( \lambda - 1 \right) + 2\left( 2\lambda - 5 \right) + 3\left( 3\lambda - 1 \right) = 0\]

\[ \Rightarrow \lambda = 1\]

Substituting  \[\lambda = 1\]   in (1),

we get the position vector of L as \[\hat{i}  + 3 \hat{j} + 5 \hat{k}\] 

Substituting  \[\lambda = 1\] in (2),

we get  , \[\overrightarrow{PL} = - 3 \hat{j} + 2 \hat{k}\]

\[= \sqrt{\left( - 3 \right)^2 + 2^2}\]

\[ = \sqrt{13}\]

Hence, the length of the perpendicular from point P on PL is \[\sqrt{13} \text{ units} \] .