It should be noted that, even though the definitions of the dot product given above are correct, the angle, theta, is the angle between the two vectors when they are attached initial point to initial point.

Also, a vector noted with ||'s on it means the magnitude or absolute value of that vector. It can be found using the length formula of: |a| = sqroot(a1^2+a2^2+...+aN^2)

In sum,

a.b = |a||b|cos(t), or a.b = a1b1 + a2b2 + ... + anbn

where a and b are two vectors in R^n and t is the angle enclosed when the two vectors are attached initial point to initial point.