Parametric Line Questions
The line intersects the plane at O, (5/2, 1/2)
What if we never find a solution? If a line is parallel to the plane, then maybe whatnhappens is that when we plug in the positions of a moving point, we actually get something that never equals 7. Because maybe we actually get a constant. Say that we have gotten 13. Well, when is 13 equal to 7? The answer is never. So that’s what would tell you that the line is parallel to the plane. You wouldn’t find a solution to the equation you get at the end. When you plug these things, you get a quantity of form something times t plus constant. This is because you are plugging in the equivalent of a plane. So in general, you have an equation of a form something times t plus something equals something. Eventually, it will have exactly one solution. The special case is when thisncoefficient of t turns out to be 0 in the end; this will happen exactly when the line is either parallel or in the plane. If you carefully consider the definition of the coefficient of t, which can be found here, you’ll find that it is equal to 1 times 2 plus 2 times 1 plus 4 times (-3). This is simplynthe dot product between the normal vector of a plane and the vector along the line. The coefficient of perpendicularity becomes zero exactly when the line is perpendicular to the normal vector. That means it’s parallel to the plane. So everything makes sense.
