Examples involving the dot product:

(1) Calculate A ⋅ B with

A = 5î + 2ĵ −k
B = 3î − 3ĵ + 3k

answer:

A ⋅ B = (5)(3) + (2)(−3) + (−1)(+3)
A ⋅ B = 15 − 6 − 3 = 6

(2) Find the angle between the above vectors.

answer:

We know their dot product, so we just need to calculate their magnitudes = =
= = = 5.48

=
= = = 5.2

Finally

A ⋅ B = cos 𝜃

𝜃 = cos⁻¹

𝜃 = cos⁻¹
𝜃 = cos⁻¹(0.21) = 77.9^o