Let $P$ be the point $(3,1,-2)$ and $L$ be the line given by $x=-4+2t$, $y=2+2t$, $z=1+t$. Find parametric equations for the line passing through $P$ and intersecting $L$ at a right angle.
[Math] Find Parametric Equations for a line passing through point and intersecting line at 90 degrees
calculusparametric
Best Answer
We know that the new line intersects $L$ at some point, so let this point be $Q=(-4+2d,2+2d,1+d)$. Since we intersect at $90^\circ$, if we take the dot product of the vector from $P$ to $Q$ and a vector parallel to $L$, we should get $0$. With this information we can solve for $d$.
We have $$\begin{align}0&=(-7+2d, 1+2d,3+d)\cdot (2,2,1) \\&=-14+4d+2+4d+3+d \\&=-9+9d \end{align}$$ Hence $d=1$.
Now we have two points on the line we are looking for: $P=(3,1,-2)$ and $Q=(-2,4,2)$, and this is enough to determine a valid parameterization (we let $t=0$ correspond to $P$ and $t=1$ correspond to $Q$): $$x=3-5t,\hspace{5mm}y=1+3t,\hspace{5mm}z=-2+4t$$