Parametric Line Questions

Parametric Line Questions 𝑇ℎ𝑒 𝑙𝑖𝑛𝑒 𝑖𝑛𝑡𝑒𝑟𝑠𝑒𝑐𝑡𝑠 𝑡ℎ𝑒 𝑝𝑙𝑎𝑛𝑒 𝑎𝑡 0, 52 , 12 ( ) What if we never find a solution? If a line is parallel to the plane, then maybe whathappens is that when we plug in the positions of a moving point, we actually getsomething that never equals 7. Because maybe we actually get a constant. Say thatwe have gotten 13. Well, when is 13 equal to 7? The answer is never. So that's whatwould tell you that the line is parallel to the plane. You wouldn't find a solution to theequation you get at the end. When you plug these things, you get a quantity of form something times t plus aconstant. This is because you are plugging in the equivalent of a plane. So ingeneral, you have an equation of a form something times t plus something equalssomething. Eventually, it will have exactly one solution. The special case is when thiscoefficient of t turns out to be 0 in the end; this will happen exactly when the line iseither 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 simplythe 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 isperpendicular to the normal vector. That means it's parallel to the plane. Soeverything makes sense.