flyingpig said:I am saying that
[PLAIN]http://img34.imageshack.us/img34/9647/unledseh.jpg
A plane's direction is determined by it's normal vector right? So if it is perpendicular to the vector <2,-1,5>, why would its normal vector be the same as <2,-1,5>?
flyingpig said:But <2,-1,5> is just a scalar multiple of <2,-1,5> (scalar = 1), that is parallel?
A tangent plane is a flat surface that touches a curved surface at a single point. It is used to approximate the behavior of a curved surface at that specific point.
The tangent plane is orthogonal, or perpendicular, to the normal vector of the curved surface at the point of contact. This is because the normal vector represents the direction of steepest ascent on the curved surface, and the tangent plane must be perpendicular to this direction in order to approximate the behavior of the surface accurately.
The tangent plane and a parallel plane are different in that the tangent plane touches the curved surface at a single point, while a parallel plane does not touch the curved surface at all. Additionally, the tangent plane is always perpendicular to the normal vector of the curved surface, whereas a parallel plane can be positioned at any angle.
No, the tangent plane can never be parallel to the curved surface. This is because the tangent plane is always perpendicular to the normal vector of the curved surface at the point of contact.
In calculus, the tangent plane is used to approximate the behavior of a curved surface at a specific point. This is important in finding the slope of a curve, calculating derivatives, and solving optimization problems.