Adding vectors analytically

To add vectors, add their components: if R = A + B, then Rx = Ax + Bx, Ry = Ay + By (and Rz = Az + Bz). The same works for any number of vectors and for subtraction.

For two vectors of magnitudes A and B with angle θ between them, R² = A² + B² + 2AB cos θ (the law of cosines).

The direction of R from A is given by tan α = B sin θ / (A + B cos θ); the law of sines, R/sin θ = A/sin β = B/sin α, relates the sides and angles of the triangle.

The resultant lies between |A − B| (θ = 180°) and A + B (θ = 0°); at θ = 90° it is √(A² + B²).

40 m and 30 m displacements at 60° to each other give R = √(1600 + 900 + 1200) ≈ 60.8 m, at about 25° to the 40 m leg.

Worked with components, a boat heading north across a current adds its velocity through the water to the water's velocity; the ground velocity is the vector sum.