Originally Posted by
HallsofIvy No, neither <4, -1, 1>.<4, 1, -1> nor <-4, -1, 1>.<4, 1, -1> is equal to 0.
What is <1, -2, 2>.<4, 1, -1>?
What is <-3, 2, -10>.<4, 1, -1>?
A line lies in the plane if and only if (x, y, z) satisifes the equation of the plane for all t. That is, put the expressions for x, y, and z in the line, in terms of t, into the equation of the plane. If the entire line is in the plane, the "t" terms will cancel out, giving a equation that is always true, like "0 = 0". If a line is parallel to the plane but not in it, the "t" terms will also cancel but give an equation that is always false, like "1= 0".