Dot product multiplication technique

The most common way to calculate the
dot product of two vectors is to multiply
their corresponding components and
sum the results. For twO vectors,
d= (aj, a,, .., a,) and b= (b,, b,.., bn)
the dot product is given by:
a•b=ajb, + asby t ...span, (end,pan) [spai
For example, if å = (2,3) and b = (4, 1),
then - b= (2)(4) + (3)(1) = 8+3 = 11.
Geometrically, the dot product is defined
by the magnitudes of the two vectors
and the angle between them. This
definition provides a deeper
understanding of what the dot product
ä-b= |d|b| cos(0)
|ā and |õ| are the magnitudes (or
lengths) of the vectors.