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Pure · Vectors

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Modelling with vectors

Vectors as physical quantities — displacement, velocity and force — with constant-velocity motion r = r₀ + tv, speed as the magnitude of velocity, bearings from north, resultant forces and equilibrium, and "passes through O".

Pure · Vectors

Modelling with vectors

Vectors as physical quantities — displacement, velocity and force — with constant-velocity motion r = r₀ + tv, speed as the magnitude of velocity, bearings from north, resultant forces and equilibrium, and "passes through O".

Why it works

Anything with a size and a direction is a vector, so vectors model the physical world: displacement (how far and which way), velocity (speed and heading), force (push and direction). In these questions you are usually told that i\mathbf{i} points east and j\mathbf{j} points north.

Constant velocity. If a particle starts at position vector r0\mathbf{r}_0 and moves with constant velocity v\mathbf{v}, then after time tt its position is r=r0+t v.\mathbf{r} = \mathbf{r}_0 + t\,\mathbf{v}. Over an interval, velocity = displacement ÷ time, where the displacement is "final position minus initial position", AB⃗=b−a\vec{AB} = \mathbf{b} - \mathbf{a}.

Speed is a magnitude. Velocity is a vector; speed is its size, speed=∣v∣\text{speed} = |\mathbf{v}|. A question asking for speed (a single number) wants the magnitude, not the velocity vector — this is the most common slip.

Bearings are measured clockwise from north as a three-figure angle. With i\mathbf{i} east and j\mathbf{j} north, a heading xi+yjx\mathbf{i} + y\mathbf{j} in the first quadrant has bearing θ\theta where tan⁡θ=xy\tan\theta = \dfrac{x}{y} (east over north) — sketch it to get the quadrant right.

Forces add as vectors. The single force with the same effect as several forces is their resultant, the vector sum F1+F2+⋯\mathbf{F}_1 + \mathbf{F}_2 + \cdots. A particle is in equilibrium when the resultant is the zero vector, so a balancing force is −(resultant)-(\text{resultant}). Its magnitude is found from components — you cannot just add the sizes of the separate forces.

"Passes through OO". A particle moving in a straight line passes through the origin exactly when OO lies on its path — that is, when its position vectors at two times are parallel (so the start, OO and a later point are collinear).