Vectors and Parametric Equations
PhD. in Mathematics
Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.
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.