a (u · v) = a |u| |v| cos θ
Further rearrangements lead to:
= | au | |v| cos θ = (au) · v = u · (av)
You may be somewhat fuzzy about how the cosine function behaves. Rather than memorize abstract stuff, visualize the unit circle with its radius projected onto the x-axis. From the picture,
the cosine of 0° = 1.0,
the cosine of 30° = 0.866,
the cosine of 45° = 0.707,
the cosine of 60° = 0.500,
the cosine of 90° = 0.0.
u · v = |u| |v| cos θ
Two vectors are oriented at 90° to each other. What is their dot product?