The equation (p⃗−p′⃗)⋅N⃗=0 basically means the set of all points that are orthogonal to the normal vector, which is a plane.
DavidVakshlyak
I think any easier way to think of it is P={x:aTx=b} where a is the normal vector, b encapsulates some information about distance from the origin. Expanding the inner product basically gives you the equation on the right.
killawhale2
@kingdish A more refined statement would be "The set of all points where any line in this set is orthogonal to the normal vector N".
jchen12197
To recap what was mentioned in lecture, if you take the dot product of a vector between any two points on a plane with the plane's normal vector, you get 0 since the two vectors are perpendicular. So, if you perform the equation with a point p that is not on the plane, the resulting dot product will not be 0 since the two vectors would not be perpendicular.
I believe p' is "any point on the plane".
The equation (p⃗−p′⃗)⋅N⃗=0 basically means the set of all points that are orthogonal to the normal vector, which is a plane.
I think any easier way to think of it is P={x:aTx=b} where a is the normal vector, b encapsulates some information about distance from the origin. Expanding the inner product basically gives you the equation on the right.
@kingdish A more refined statement would be "The set of all points where any line in this set is orthogonal to the normal vector N".
To recap what was mentioned in lecture, if you take the dot product of a vector between any two points on a plane with the plane's normal vector, you get 0 since the two vectors are perpendicular. So, if you perform the equation with a point p that is not on the plane, the resulting dot product will not be 0 since the two vectors would not be perpendicular.