Find the equation of the perpendicular

Question

Find the equation of the perpendicular drawn from the point P (2,4,-1) to the line?

$\frac{x+5}{1} =\frac{y+3}{4}= \frac{z-6}{-9}$

Answer 1

Let $A(-5+t,-3+4t,6-9t)$ be the point in the given line such that $\vec{AP}\perp\vec{(1,4,-9)}$.

Hence, $\vec{PA}(-7+t,-7+4t,7-9t)$ and $$(-7+t,-7+4t,7-9t)(1,4,-9)=0$$ or $$1(-7+t)+4(-7+4t)-9(7-9t)=0,$$ which gives $t=1$ and $\vec{PA}(-6,-3,-2)$, which gives the answer: $$\{(2,4,-1)+t(6,3,2)|,t\in\mathbb R\}$$ or in your form: $$\frac{x-2}{6}=\frac{y-4}{3}=\frac{z+1}{2}$$

Answer 2

Hint:

Find the plane which is orthogonal to the given line and passes through $P$. Then find the intersection of the given line and the plane. The required line is the line passing through the two points.