Scalars and Vectors — Physical Quantities in a Plane

When motion was confined to a straight line, direction could be captured with nothing more

than a plus or minus sign. Once a particle is free to move anywhere in a plane, a single

sign is no longer enough — direction itself becomes a genuinely two-dimensional idea, and

every physical quantity we work with falls into one of two families.

A scalar quantity is completely specified by a magnitude (a number) together with an

appropriate unit. Mass, distance, time, speed, temperature, work, and energy are all

scalars. Scalars of the same kind combine by ordinary arithmetic: a mass of 2 kg added to a

mass of 3 kg is simply 5 kg, with no direction to account for.

A vector quantity needs both a magnitude and a direction to be fully specified, and it

must obey the special rules of vector algebra (addition, subtraction, and the two kinds of

vector multiplication) developed through this chapter — ordinary arithmetic on the numbers

alone is not enough. Displacement, velocity, acceleration, force, and momentum are all

vectors: knowing that a car moved "5 km" tells you nothing about where it ended up unless

you also know the direction it travelled in.

A vector is written with an arrow over the symbol, $\vec A$, and its magnitude (always a

non-negative number, with the vector's unit) is written $|\vec A|$ or simply $A$.

Geometrically a vector is drawn as an arrow: the length of the arrow (to some chosen scale)

represents the magnitude, and the arrowhead represents the direction.

This chapter builds up the full vector toolkit needed for two-dimensional motion — position

and displacement vectors, equality of vectors, scaling a vector by a real number, adding and

subtracting vectors, resolving a vector into components, the two ways of multiplying two

vectors together (the scalar and vector products), relative velocity, and finally the

kinematics of motion in a plane under constant acceleration, culminating in projectile

motion and uniform circular motion — the two most important two-dimensional motions in the

WBCHSE syllabus.