The Vector Dot Product in Two Dimensions

Show that the dot product of two unit vectors:

A = ( a , b )  and  C = ( c , d )

is ac+bd.

Given both vectors are unit vectors:

a2 + b2 = c2 + d2 = 1

The rotation of C by 90° is R = ( -d , c )

Vector diagram showing relationship between vectors A, C, and their rotated components forming a right triangle.

A + mR = lC

Where l is the cosine of the angle between A and C, and m is the sine:

a - md = lc ad - lcd = m (1) b + mc = ld -bc + ldc = m (2)

Equate (1) and (2):

ad + bc = ldc + lcd

Multiply by cd:

ca + bd = l ( d2 + c2 )

Since c2+d2=1:

l = ac + bd

Therefore, the dot product A C = ac+bd as was to be shown.