Vectors and Planes - Concept
An important calculation when dealing with vectors and planes, is being able to find a vector normal to a plane through a specific point. There are methods for finding the normal or perpendicular vector to a plane and finding the plane to which a vector is normal. This relationship between vectors and planes is important and is used in the calculation of the angle between planes.
It's really easy to get the equation of a plane if you know two things, first a point that lies in the plane and second a vector that's normal to the plane. And let me define that when we say a vector n is normal or perpendicular to plane m we mean that n is perpendicular to any vector in plane m. So if you look at the picture I've drawn 2 vectors v and u and this vector n if it's normal to plane m it'll be perpendicular to each of these 2 guys. So really easy result here if n which has components a, b, c is normal to plane m an equation for that plane is ax+by+cz=d these ab and c these coefficients are exactly the components of the normal vector n so it's really easy to come up with these part of the equation of a plane and it turns out that the converse of this is true too.
If you have an equation of a plane you can easily come up with a normal vector and that's what we're going to do now. Find the vector perpendicular to the plane, these coefficients give it to you. Vector n would be 2 negative 3 6 these coefficients here n would be, we don't have an x term so I put a 0 down for that but I'd have 5 and negative 4 and here I don't have an x or a y term so my normal vector would be 0, 0 negative 4.