Vector reflection in a plane. Find the equation of the image of a line `(x,y,z) = (1,2,3) + t(1,-1,0)` reflected in the plane `x+y+z-3=0`

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To start, let's figure out what it means to reflect the line in the plane. We can use an affine transformation matrix (see link below), but this does not illustrate what's going on conceptually. To think of this visually, remember what it means to reflect. For any point P (or its associated vector from the origin `vecP`) on the line, we take a vector normal to the plane that reaches P. We then take the negative vector from the surface of the plane to find the image of P (P' or its associated vector from the origin `vec(P')` ) based on reflection. Put another way, supoose we have a normal, `vecn` , and P is a perpendicular distance `d/|vecn|` away from a point M (represented by a vector from the origin `vecM`) on...

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