The Gradient and Directional Derivatives

Main Points
An ordered pair of real numbers is a 2-dimnesional vertor which is symbolised by a bold letter. The difference between a point and a vector is that the point is simply a point location on the plane whilst the vector is an arrow. The length of a vector (a,b) is denoted by (a, b). A unit vector has a length of one unit. For any nonzero vector (a, b), you can find a unit vector in the same direction by dividing each coordinate of (a, b) by the length of the vector.
The dot product of two vector is as follows: (a1, a2) · (b1, b2) = a1b1 + a2b2; the result is a single number.
The partial derivative fx(x, y) of a function z = f(x, y), tells us the rate of change in the x-direction just as fy(x, y) tells us the rate of change in the y-direction. A gradient seems to have the following properties:
The gradient vector always points in the direction of greatest increase.
The gradient vector is always perpendicular to the tangent of the curve at which it is rooted.
The opposite direction to that of the gradient is the direction of greatest decrease.

Challeneges
After looking at the properties of a gradient, I dont think I understand exactly how we got to that point because I do not understand wy we have those properties.

Reflections
It looks to me as though vectors might be used to simplify things like a contour diagram to facilitate understanding for those less familiar with the diagram. And they are more useful than just points here, especially since they have a directional factor as well.