Adding and subtracting vectors graphically

Triangle (head-to-tail) method: place the tail of B at the head of A; the arrow from the tail of A to the head of B is R = A + B.

Vector addition is commutative, A + B = B + A, and associative, (A + B) + C = A + (B + C).

Adding A and −A gives the null (zero) vector 0, of zero magnitude and no fixed direction. A + 0 = A, λ0 = 0 and 0A = 0. A displacement that returns to its start is the null vector.

Subtraction is addition of the reversed vector: A − B = A + (−B).

Parallelogram method: draw A and B from a common tail and complete the parallelogram; the diagonal from that tail is A + B. It gives the same result as the triangle method.

Rain falling vertically at 8 m/s in a wind blowing from east to west at 6 m/s moves at 10 m/s, at an angle whose tangent is 6/8 (about 37°) from the vertical, drifting west. To meet it, the umbrella is tilted about 37° from the vertical towards the east.