 Hello, and welcome to this screencast on section 9.5, Lines and Planes in Space. This screencast is going to cover Planes in Space. Jumping right in, a plane P in space is the set of all terminal points of vectors emanating from a given point p-naught perpendicular to a fixed vector in. As we have pictured here, we see that the point labeled p is the terminal point of this vector, which is perpendicular to this given vector in. We can choose any other point on this plane shaded in gray, say over here, and this will be the terminal point of some vector that is again perpendicular to n. This picture is helpful to visualize what's happening, but what does the equation of the plane actually look like? Well, since n and the vector from p-naught to p are perpendicular, we know that the dot product of these two vectors must be zero. This immediately gives us the vector equation of a plane. So this is one way we can describe a plane. We can actually take this one step further to get another representation for the equation of a plane. Plugging in the components of the vector in, which are a, b, and c, and the components of the vector from p-naught to p, which are given by v's, and taking the dot product, we see that we get an equation for the plane in terms of x, y, and z. This equation is called the scalar equation of a plane, where we have zero is equal to this expression. To summarize our work from the previous slide, we have the following two ways to represent the equation of a plane. At the top, we have the scalar equation, and at the bottom, we have the vector equation. One thing to note is that if we have a vector that's perpendicular to a plane and any point on the plane, then we can use either one of these representations to immediately find the equation of the plane.