vector addition

Meaning: Vectors such as displacement, velocity, acceleration and force have magnitude and direction, so they combine by geometry, not plain arithmetic. Graphically, place the tail of the second vector at the head of the first; the resultant runs from the first tail to the last head, and the order does not matter. Analytically, resolve each vector into x and y components, add each axis separately, then recombine. Parallel vectors add in magnitude, antiparallel ones subtract, and perpendicular ones give a Pythagorean resultant.

Equation: Rx = Ax + Bx, Ry = Ay + By, |R| = √(Rx² + Ry²), θ = tan⁻¹(Ry/Rx), where R is the resultant, A and B are the vectors being added, subscripts mark components, and θ is the resultant's angle from the x axis. Units are those of the quantity: m, m/s or N.

On the MCAT: Typical setups are a boat crossing a river current, a plane flying in wind, or forces on a block on an incline. You find a net force, a resultant velocity, or the condition for equilibrium (components sum to zero). Right-triangle estimates such as 3-4-5 are common.

Common confusion: Students add magnitudes directly. The magnitude of the resultant of A and B ranges from |A − B| to A + B, and it reaches the maximum only when the vectors are parallel.