How do we prove these properties of the inner product?  Let’s start with the fact that orthogonal vectors have 0 inner product.  Suppose one vector is (x,y), and WLOG x,y>0.  Then, if we rotate that by 90 degrees counterclockwise, we’ll get (y, -x).  Rotating the vector is just like rotating the coordinate system in the opposite direction.  And (x,y)*(y,-x) = xy – yx = 0.
Next, note that v*w = (v*w)/(||v||||w||)  * ||v||||w||  This means that if we can show that when v and w are unit vectors v*w = cos alpha, then it will follow that in general v*w = ||v|| ||w|| cos alpha.  So suppose v and w are unit vectors.
Next, note that if w1 + w2 = w, then v*w = v*(w1+w2) = v*w1 + v*w2.  For any w, we can write it as the sum of w1+w2, where w1 is perpendicular to v, and w2 is in the same direction as v.  So v*w1 = 0.  v*w2 = ||w2||, since v*w2/||w2|| = 1.  Then, if we just draw a picture, we can see that cos alpha = ||w2|| = v*w2 = v*w.