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Overview
Scalars and vectors are two major categories of physical quantities. A scalar has magnitude only. A vector has magnitude and direction. This difference is central to mechanics, forces, motion, fields, momentum, and many other areas of physics.
Understanding the distinction prevents major errors. A speed of 20 meters per second and a velocity of 20 meters per second east are not the same statement, because velocity includes direction.
Scalars
A scalar is a quantity described by size or amount. Common scalar quantities include mass, time, temperature, energy, distance, speed, volume, pressure, and electric charge.
Scalars can usually be added by ordinary arithmetic when they represent the same type of quantity and use compatible units.
Vectors
A vector is a quantity described by both magnitude and direction. Common vector quantities include displacement, velocity, acceleration, force, momentum, electric field, and magnetic field.
Vectors require directional reasoning. Two forces of equal size can cancel if they act in opposite directions or combine if they act in the same direction.
Representing Vectors
Vectors can be represented with arrows. The length of the arrow represents magnitude, and the arrow direction represents the vector direction. Vectors can also be represented with components, such as horizontal and vertical parts.
Coordinate systems make vector representation more precise. Once directions are assigned to axes, vectors can be broken into components and handled with algebra.
Vector Components
A vector component is the part of a vector that lies along a chosen axis. In two-dimensional problems, a vector is often broken into horizontal and vertical components.
Components allow complex vector problems to be solved by separating directions. The horizontal and vertical parts can be analyzed independently when the model allows it.
Common Misunderstandings
A common mistake is treating vectors as if they were scalars. For example, adding two velocities without considering direction can produce a wrong answer.
Another mistake is confusing distance with displacement. Distance is scalar and measures total path length. Displacement is vector and measures change in position from start to finish.