Norm

( |x| = \sqrt{x_1^2+…+x_n^2} ) in n-dimensional Euclidean space
( |x|1 = \sum\limits{i=1}^n |x_i| )
( |x|p = (\sum\limits{i=1}^n |x_i|^p)^{1/p} )

Vector dot product

( \vec{a} \cdot \vec{b} = \sum\limits_{i=1}^n a_i b_i = |\vec{a}| |\vec{b}| cos(\theta) )
( cos(\theta) = \frac{\vec{a} \cdot \vec{b}} {|\vec{a}| |\vec{b}|} )
And dot product also means projection =>
( |\vec{a}| cos(\theta) ) is the component of vector a in direction b. (project a to b)
( |\vec{b}| ) is the length of vector b. However, if b is normal vector, then ( |\vec{b}| ) is one.