Components and adding vectors analytically

A vector at angle θ to the x-axis can be split into two components at right angles: Aₓ = A cos θ and A_y = A sin θ. A handy rule: the part next to the angle takes cos, the other part takes sin. So A = Aₓ î + A_y ĵ.

Example: an aeroplane climbs at 30° to the ground, and its speed along the ground is 250 km/h. Then 250 = v cos 30°, so v = 250/0.866 ≈ 289 km/h, and it rises at v sin 30° ≈ 144 km/h.

Going back: the size is A = √(Aₓ² + A_y²) (add A_z² in three dimensions) and the direction is tan θ = A_y/Aₓ.

To add vectors, add their components: Rₓ = Aₓ + Bₓ, R_y = A_y + B_y, then rebuild R. This works for any number of vectors, so it is the best method for three or four forces. (î + 4ĵ – 2k̂) + (3î – 5ĵ + k̂) = 4î – ĵ – k̂, of size √18.

For two vectors at angle θ there is also a formula (the parallelogram law worked out):

• R² = A² + B² + 2AB cos θ, and the resultant makes angle α with A, where tan α = B sin θ/(A + B cos θ).
• For A – B, change +2AB cos θ to –2AB cos θ.

Example: forces of 5 N and 7 N at 60°. R² = 25 + 49 + 2 × 5 × 7 × 0.5 = 109, so R ≈ 10.4 N, at tan α = 6.06/8.5, about 35.5° from the 5 N force. Check the ends: at θ = 0° the formula gives A + B, and at 180° it gives A – B.

To give a vector a chosen size in a chosen direction, multiply that size by the unit vector. A vector of size 26 along 5î + 12ĵ (whose size is 13) is 26 × (5î + 12ĵ)/13 = 10î + 24ĵ.

The component of A along any direction n̂ (a unit vector) is A · n̂ = Aₓnₓ + A_yn_y. The component of 4î + 2ĵ along î + ĵ is (4 + 2)/√2 = 3√2 ≈ 4.2.