Mr. Integral
Math notes/Calculus III/Vectors and Geometry in Space

Vectors, Dot Products and Cross Products — Cheat sheet

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Core rule

u⋅v=∥u∥∥v∥cos⁡θu\cdot v=\|u\|\|v\|\cos\theta, ∥u×v∥=∥u∥∥v∥sin⁡θ\|u\times v\|=\|u\|\|v\|\sin\theta, and volume is ∣u⋅(v×w)∣|u\cdot(v\times w)|.

Watch for

The dot product is a scalar; the cross product is an oriented vector. A zero vector has no direction.