Reflecting a line from plane

Question

I have a plane given by equation $0=10x+2y-3z$ and a line described by vector with variable $t\ge0$

$(1-t, 1+2t, 1+t)$

How to calculate reflected vector of this line from plane?

We treat line as if it was light reflecting from surface.

Answer 1

1) First calculate the unit vector $\hat{\boldsymbol{n}}$ in the direction of your line, by choosing a suitable value for $t$ and then normalizing.

2) Calculate the components of $\hat{\boldsymbol{n}}$ that are perpendicular ($\hat{\boldsymbol{n}}_{\perp}$) and parallel ($\hat{\boldsymbol{n}}_{||}$) to the plane. Thus, $\hat{\boldsymbol{n}} = \hat{\boldsymbol{n}}_{\perp} + \hat{\boldsymbol{n}}_{||}$.

3) The reflection is then generated by the unit vector $\hat{\boldsymbol{n}}' = -\hat{\boldsymbol{n}}_{\perp} + \hat{\boldsymbol{n}}_{||}$.