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## Main Question or Discussion Point

I take 2 points given by the vectors of coordinates ##\vec{p}_i,\vec{p}_j## and a plane spanned by ##\vec{e}_k,k=1,2##.

All the vectors are in dimension n.

I want to find the intersection of the segment described by the extremities given by the ##\vec{p}_k## with the plane, if any.

Is it correct to say the following :

Let $$\vec{d}_k=\vec{p}_k-Proj(\vec{p}_k,\pi_{e_1,e_2}),k=i,j$$.

if ##\vec{d}_i\cdot\vec{d}_j<0## there is an intersection ?

The intersection point is found by selecting 3 equations among the system : ##\alpha(\vec{p}_j-\vec{p}_i)+\vec{p}_i=\beta\vec{e}_1+\gamma\vec{e}_2## for which the determinant is nonzero and solving for the greek coefficients ?

All the vectors are in dimension n.

I want to find the intersection of the segment described by the extremities given by the ##\vec{p}_k## with the plane, if any.

Is it correct to say the following :

Let $$\vec{d}_k=\vec{p}_k-Proj(\vec{p}_k,\pi_{e_1,e_2}),k=i,j$$.

if ##\vec{d}_i\cdot\vec{d}_j<0## there is an intersection ?

The intersection point is found by selecting 3 equations among the system : ##\alpha(\vec{p}_j-\vec{p}_i)+\vec{p}_i=\beta\vec{e}_1+\gamma\vec{e}_2## for which the determinant is nonzero and solving for the greek coefficients ?