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## Main Question or Discussion Point

Given are a plane E and a line l in general position. I need to find a plane that contains l and intersects E at a given angle [tex]\alpha[/tex]. All of this happens in R^3.

The interesting part is to find the normal of the unknown plane, let us call this normal x. I came up with the following equations:

[tex]x^Tx = 1[/tex]

[tex]x^Tn = \cos \alpha[/tex]

[tex]x^Td = 0[/tex]

in which n is the unit normal vector of the given plane, and d is the direction vector of l.

(1) says I want a unit vector, (2) says the planes need to intersect at a given angle, and (3) says the plane needs to be parallel to the given line.

These equations can easily be solved by writing them out in component form with x=(x1, x2, x3) and doing some substitutions, which will yield a quadratic equation in one of the parameters.

However, I am wondering if there is a more elegant way to express the solution - something more matrixy-vectory? The equations look simple enough...?

The interesting part is to find the normal of the unknown plane, let us call this normal x. I came up with the following equations:

[tex]x^Tx = 1[/tex]

[tex]x^Tn = \cos \alpha[/tex]

[tex]x^Td = 0[/tex]

in which n is the unit normal vector of the given plane, and d is the direction vector of l.

(1) says I want a unit vector, (2) says the planes need to intersect at a given angle, and (3) says the plane needs to be parallel to the given line.

These equations can easily be solved by writing them out in component form with x=(x1, x2, x3) and doing some substitutions, which will yield a quadratic equation in one of the parameters.

However, I am wondering if there is a more elegant way to express the solution - something more matrixy-vectory? The equations look simple enough...?