Linear Algebra 45, Hyperplanes and Normal vector to the plane

Linear Algebra 45, Hyperplanes and Normal vector to the plane

[salmanjavid]

Shouldn't the origin be outside the hyper plane in general case? I mean a hyper plane can be shifted from the origin. And moreover the given explanation also confuses in terms of the normal vector because if it is normal to hyper plane then it should be normal to every displacement vector in the the plane, including OU and OW (If point O is in the hyper plane) and the dot product should be zero but the fundamental equation suggests that it could be any constant 'b'. 
Correct me if I'm wrong

[gyateen]

If a vector is normal to a plane, then how can its dot product with any vector in a plane be constant. Shouldn't it be zero ??

[Kinshuk]

Great video, but I think "O" should not necessarily be on the hyperplane. It could be outside as well.

[googlesong]

It is very confusing because the hyperplane doesnt have to corss the origin. x+y+z=2 is just the plane of x+y+z=0 shifted from the origin and the normal vector (1, 1, 1) is orthogonal to the plane x+y+z = b no matter how much you shift the plane. This means b doesn't have to be 0. b can be any number and the (1, 1, 1) is orthogonal to any of these shifted planes.

[ruralmetropolitan]

Great explanation, thank you so much!!