Question

Find the equation of the perpendicular drawn from the point P (−1, 3, 2) to the line $\overrightarrow{r}=\left(2\hat{j}+3\hat{k}\right)+\lambda \left(2\hat{i}+\hat{j}+3\hat{k}\right).$ Also, find the coordinates of the foot of the perpendicular from P.

Answer 1

Let L be the foot of the perpendicular drawn from the point P ($-$1, 3, 2) to the line $\overrightarrow{r}=\left(2\hat{j}+3\hat{k}\right)+\lambda \left(2\hat{i}+\hat{j}+3\hat{k}\right).$



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

Now,

$$\begin{gathered} \overrightarrow{PL}=\mathrm{Position}\mathrm{vector}\mathrm{of}L-\mathrm{Position}\mathrm{vector}\mathrm{of}P \\ \Rightarrow \overrightarrow{PL}=\left\{2\lambda \hat{i}+\left(2+\lambda \right)\hat{j}+\left(3+3\lambda \right)\hat{k}\right\}-\left(-\hat{i}+3\hat{j}+2\hat{k}\right) \\ \Rightarrow \overrightarrow{PL}=\left(2\lambda +1\right)\hat{i}+\left(\lambda -1\right)\hat{j}+\left(3\lambda +1\right)\hat{k}...\left(2\right) \end{gathered}$$

Since $\overrightarrow{PL}$ is perpendicular to the given line, which is parallel to $\overrightarrow{b}=2\hat{i}+\hat{j}+3\hat{k}$, we have

$$\begin{gathered} \overrightarrow{PL}.\overrightarrow{b}=0 \\ \Rightarrow \left\{\left(2\lambda +1\right)\hat{i}+\left(\lambda -1\right)\hat{j}+\left(3\lambda +1\right)\hat{k}\right\}.\left(2\hat{i}+\hat{j}+3\hat{k}\right)=0 \\ \Rightarrow 2\left(2\lambda +1\right)+1\left(\lambda -1\right)+3\left(3\lambda +1\right)=0 \\ \Rightarrow \lambda =-\frac{2}{7} \end{gathered}$$

Substituting $\lambda =-\frac{2}{7}$ in (1), we get the position vector of L as $-\frac{4}{7}\hat{i}+\frac{12}{7}\hat{j}+\frac{15}{7}\hat{k}$.

So, the coordinates of the foot of the perpendicular from P to the given line is L $\left(-\frac{4}{7},\frac{12}{7},\frac{15}{7}\right)$.
Substituting $\lambda =-\frac{2}{7}$ in (2), we get
$\overrightarrow{PL}=\frac{3}{7}\hat{i}-\frac{9}{7}\hat{j}+\frac{1}{7}\hat{k}$

Equation of the perpendicular drawn from P to the given line is
$$\begin{gathered} \overrightarrow{r}=\mathrm{Position}\mathrm{vector}\mathrm{of}P+\lambda \left(\overrightarrow{PL}\right) \\ =\left(-\hat{i}+3\hat{j}+2\hat{k}\right)+\lambda \left(3\hat{i}-9\hat{j}+\hat{k}\right) \end{gathered}$$