Question

Find the vector equation of the line passing through the point whose position vector is $3\hat{i}+\hat{j}-\hat{k}$ and which is parallel to the vector $2\hat{i}-\hat{j}+2\hat{k}$. If $P$ is a point on this line such that $AP=15$, find the position vector of $P$.

Answer 1

Equation of line:$\frac{x-3}{2}=\frac{y-1}{-1}=\frac{z+1}{2}$

Let $A(3,1,-1)$

$P(2r+3,-r+1,2r-1)$

$AP=15$

$\begin{array}{c}{\left(2r\right)}^2+{\left(-r\right)}^2+{\left(2r\right)}^2=225\\ 9r^2=225\\ r^2=25\\ r\pm 5\\ P\left(13,-4,9\right)\\ P\left(-7,6,-11\right)\end{array}$