Question

Find the equation of the line parallel to $2\overline{i}-\overline{j}+2\overline{k}$ and which passes through point ${}^{\prime }{A}^{\prime }(3\overline{i}+\overline{j}-\overline{k})$. if P is a point on the line such that $AP=15$. Find the position vectors of $P$.

Answer 1

We have,

$\begin{array}{c}\overrightarrow{r}=3\hat{i}+\hat{j}-\hat{k}+\lambda \left(2\hat{i}-\hat{j}+2\hat{k}\right)\\ L:\frac{x-3}{2}=\frac{y-1}{-1}=\frac{z+1}{2}\end{array}$

Let $P=\left(2r+3,-r+1,2r-1\right)A\left(3,1,-1\right)$

$\begin{array}{c}AP=\sqrt{{\left(2r\right)}^2+{\left(-r\right)}^2+{\left(2r\right)}^2}=15\\ \Rightarrow 3\sqrt{r^2}=15\\ \Rightarrow \sqrt{r^2}=5\\ \therefore r=\pm 5\\ P=\left(13,-4,9\right)or\left(-7,6,-11\right)\end{array}$