Scalars and vectors

Every physical quantity is either a scalar or a vector.

Scalars

• Include quantities such as distance, speed, mass, and energy
• Are fully described by a magnitude alone and carry no directional information

Vectors

• Include quantities such as position, displacement, velocity, acceleration, and force
• Require both a magnitude and a direction to be fully described

This distinction determines how quantities combine: scalars add arithmetically, while vectors add component by component along each axis direction.

Vector Representation

• Vectors can be represented visually as arrows, where the arrow's length is proportional to the magnitude and the arrow's direction matches the vector's direction
• In calculations, vectors are expressed in unit vector notation using î, ĵ, and k̂ for the x-, y-, and z-directions
• The position vector r⃗ points from the origin to a point in space, while the unit position vector r̂ has a magnitude of one and carries only directional information

Resultant Vector

• A resultant vector is found by summing the components of the individual vectors along each axis
• These summed components are then combined using the Pythagorean theorem to find the magnitude and the inverse tangent to find the direction

In one-dimensional problems, opposite directions are represented by opposite algebraic signs, and the positive direction must be defined before any calculation begins.

At a Glance