✖️ Dot & Cross Product

📖 The Physics

Two vectors can be multiplied in two very different ways. The dot (scalar) product answers "how much do they point the same way?" and gives a plain number. The cross (vector) product answers "how much are they perpendicular?" and gives a new vector at right angles to both.

Dot (scalar) product, and cross (vector) product with n̂ ⟂ to both:

In the dot view, the yellow segment is the projection of Bonto A; multiply its length by |A| to get A·B. It is largest when the vectors are parallel (θ = 0, cos = 1) and zero when they are perpendicular (θ = 90°).

In the cross view, the shaded parallelogram's area equals |A×B|. It is largest when the vectors are perpendicular and zero when they are parallel. The direction (⊙ out of the page / ⊗ into the page) follows the right-hand rule.