u = 9 i - 6 j
v = -3 i - 2j
w = 19 i + 15 j
u • v = (9 i - 6 j) • (-3 i - 2j)
Distribute the dot products:
u • v = 9*(-3) (i • i) + 9*(-2) (i • j) + (-6)*(-3) (j • i) + (-6)*(-2) (j • j)
i and j are orthogonal unit vectors, so their dot products are 0, while i • i = j • j = 1. So we have
u • v = 9*(-3) + (-6)*(-2) = -27 + 12 = -15
In other words, the dot product can be computed by simply multiplying corresponding components, and taking the total.
u • w = 9*19 + (-6)*15 = 81