How:

(2,3) • (5,7) = 2·5 + 3·7 = 31

(9,4) • (-2,6) = 9·(-2) + 4·6 = 6

(9,4,3) • (-2,6,5) = 9·(-2) + 4·6 + 3·5 = 21

(2,3,4) • (5,7,-8) = 2·5 + 3·7 + 4·(-8) = -1

(9,4)² = (9,4)•(9,4) = 9·9 + 4·4 = 97

(-2,6)² = (-2,6)•(-2,6) = (-2)·(-2) + 6·6 = 40

Why:

If you move into a certain direction, you can express this as a **vector**

North 4m/s, East 6m/s (6,4)

North 1m/s, West 3m/s, Up 5m/s (-3,1,5)

Sometimes, it is helpful to determine the **angle** between **two vectors**. This can be done with the **cosine law** in combination with vectors, which leads you to the dot product

(9,4) • (-2,6) 6

∡<(2,3), (5,7)> = arccos --------------- = arccos ------- ≈ 84.5°

√(9,4)²·√(-2,6)² √97·√40

It is easier to determine if two vectors are **orthogonal/perpendicular** to each other, since arccos(0) = 0° you only need to check the dot product

(2,3)•(5,7) = 31 not orthogonal

(2,3)•(-9,6) = 0 orthogonal

The calculations above also work for three-dimensional including the z-axis