Measurement and vectors formulas

A = size of the vector. Flat (2-D) problem: keep A_z = 0. Add or subtract vectors right in the boxes, part by part: for C = A − B type 10.4−6.25 in the Aₓ box (same for y). 2A + B: type 2×10.4+6.25. Vectors given as size + angle: type the parts, e.g. Aₓ box: 175cos(60)+80cos(90), Aᵧ box: 175sin(60)+80sin(90). Boat and river, plane and wind, rain seen from a car: add the velocity parts the same way.

θ = direction, counterclockwise from the +x axis (east). If Aₓ < 0: add 180°. If Aₓ > 0 and Aᵧ < 0: add 360°. Aₓ = 0: θ = 90° (Aᵧ > 0) or 270° (Aᵧ < 0). Angle from vertical or like 59° W of N: vertical (N) part in Aₓ, other part in Aᵧ, both positive. Right triangle: θ is the corner next to side Aₓ. Empty Aᵧ gives the far side (river width). Sagging rope: Aᵧ = sag, Aₓ = half the span. Radians: answer × π/180.

θ is measured from the +x axis. Angle given from the y axis: type 90−angle. Angle in radians: type it ×180/π. Ramp: A = m×9.8, θ = ramp angle gives Aₓ = normal force N. Cross a river straight: A = boat speed, Aₓ = river speed gives θ = aim angle upstream from the bank. 3-D, angle from +z given: type it as θ to get A_z. For Aₓ and Aᵧ type A×sin(z angle) in the A box. Empty θ gives 0° to 180°; if Aᵧ < 0 use 360° − θ.

θ is measured from the +x axis. Ramp: A = m×9.8, θ = ramp angle gives the weight part down the ramp. Sagging rope: first find θ = tan⁻¹(sag/half span). Then A = tension in one half, Aᵧ = m×9.8/2. Two cables at 90° to each other holding a weight: A = m×9.8, θ = one cable's angle above level, Aᵧ = that cable's tension. Swim straight across a river: A = swim speed, Aᵧ = river speed, θ = aim angle from straight across.

φ = angle between A and B (put their tails together). From parts: type AₓBₓ+AᵧBᵧ+A_zB_z in the A·B box, e.g. 2×(−3)+5×0+0×4. Sizes from parts: type A as √(2²+5²+0²). A·B = 0: the vectors are at 90°. A·B < 0: φ is over 90°.

φ = angle between A and B (tails together). Empty φ gives two angles that both work. Direction: right-hand rule. Fingers along A, curl toward B, thumb = A×B. B×A points the opposite way. x×y = z, y×z = x, z×x = y. Parts by hand: x = AᵧB_z − A_zBᵧ, y = A_zBₓ − AₓB_z, z = AₓBᵧ − AᵧBₓ. Parallel vectors: A×B = 0.

Type kg and m³ (answer in kg/m³). For g/cm³ divide the answer by 1000. 1 g = 0.001 kg. 1 L = 0.001 m³. 1 cm³ = 1E−6 m³. Volume: cube the factor. 1 ft³ = 0.3048^3 m³. Sphere: V = 4/3×π×(d/2)^3. Box: V = length×width×height. Unit changes: 1 in = 2.54 cm, 1 ft = 0.3048 m, 1 mi = 1609 m, 1 h = 3600 s, 1 yr = 365 days. Prefixes: k = 1E3, M = 1E6, c = 0.01, m = 0.001, μ = 1E−6, n = 1E−9.