vector

Meaning: A vector has magnitude and direction. A scalar has magnitude only. Displacement, velocity, acceleration, force, momentum and torque are vectors; distance, speed, mass, energy, work and temperature are scalars. Vectors add head-to-tail, or by adding x and y components separately, so the result depends on direction.

Equation: Ax = A cos θ, Ay = A sin θ, A = √(Ax² + Ay²). A is the magnitude (in the unit of the quantity, such as m for displacement), θ is the angle from the +x axis (rad), and Ax and Ay are the components along x and y (same unit as A).

On the MCAT: Expect forces resolved on an incline, projectile motion with independent horizontal and vertical parts, relative velocity (a boat in a current), and equilibrium problems where components sum to zero. Torque and magnetic force use cross products with the right-hand rule; work is a dot product and is a scalar.

Common confusion: Students add magnitudes directly. Two 3 N forces at right angles give about 4.2 N, not 6 N; magnitudes add only for parallel vectors. Also, a round trip has zero displacement and zero average velocity but nonzero distance and speed. A minus sign shows direction, not smaller size.