I've found this challenge to solve that I have no idea where to start. We are supposed to find the symmetric line to this one
$r: (10,1,2) + \lambda (3,1,1)$
with regard to the plane
$\alpha \ \{ 3x+y-z-2 = 0$
Where should I begin? The info that I've extracted are:
Intersection point of $r$ and $\alpha$: $(1,-2,-1)$
Vector of $\alpha$: (3,1,-1)
Vector of line $r$: $(3,1,1)$
Thank you.
HINT: Your line will be of the form $$\mathbf r_{\text{reflect}}(t) = \mathbf vt+\mathbf b$$
Because you found the intersection of $\mathbf r$ and $\mathbf \alpha$, you can use that for your postion vector $\mathbf b$. Why? To find $\mathbf v$ just decompose the direction vector of $\mathbf r$, $(3,1,1)$, into a vector parallel to the plane and a vector perpendicular to the plane. $$(3,1,1) = \mathbf u = \mathbf u_\| + \mathbf u_\perp$$
$\mathbf v$ will simply be $\mathbf v = \mathbf u_\| - \mathbf u_\perp$. Why?