The dot product commutes and distributes, but never try to "cancel a vector" from both sides, because this isn't ordinary multiplication.
Module goal. Introduce the operation that multiplies two vectors into a number, capturing the angle between them, projection, and when they're perpendicular.
eastmath.com · 27.4 The Dot Product: Angles and Perpendicularity · 27.4.7 The Rules, and a Word of Caution