u = 2i - 6j
v = 3i + 9j
Take the dot product:
u • v = (2i - 6j) • (3i + 9j)
u • v = (2i • 3i) + (2i • 9j) + (-6j • 3i) + (-6j • 9j)
u • v = 6 (i • i) + 18 (i • j) - 18 (j • i) - 54 (j • j)
The dot product is defined so that for the orthogonal unit vectors i and j, we have i • i = j • j = 1, and i • j = 0. So the above reduces to
u • v = 6 + 0 - 0 - 54
u • v = -48
