Adding vectors

Triangle law: to add a and b, draw b starting at the head of a. The vector from the tail of a to the head of b is a + b. A trip from A to B and then B to C has the same net effect as AC, so AB + BC = AC.

Going round a closed triangle gives nothing: AB + BC + CA = 0, since the path ends where it began.

Subtraction is adding the negative: a − b = a + (−b). For example AC − BC = AC + CB = AB.

Parallelogram law: draw a and b from one point and complete the parallelogram; the diagonal from that point is a + b. It gives the same vector as the triangle law.

Addition is commutative, a + b = b + a, and associative, (a + b) + c = a + (b + c), so a sum of several vectors can be written without brackets. Also a + 0 = a and a + (−a) = 0.

Example with numbers: a drone flies at 4 m/s due north through air that drifts 3 m/s due east. Its velocity over the ground is 3î + 4ĵ, of speed √(9 + 16) = 5 m/s, heading tan⁻¹(3/4) ≈ 36.9° east of north.