What is a vector?

Some quantities need only a number: 5 kg, 30 seconds, 12 km. These are scalars: they have size only.

Others make sense only with a direction too. '10 km' does not tell you where you ended up; '10 km north' does. These are vectors: they have size and direction. Displacement, velocity, force and acceleration are vectors; distance, speed, mass, time and temperature are scalars.

A vector is drawn as an arrow. The arrow from point A to point B is written AB (with an arrow on top). A is the initial point, B the terminal point. The length of the arrow is the vector's magnitude, written |AB|.

Types of vector to know by name:

• Zero vector: start and end at the same point. Length 0, no fixed direction.
• Unit vector: length exactly 1.
• Coinitial vectors: they all start at the same point.
• Collinear vectors: parallel to the same line, whatever their lengths or senses.
• Equal vectors: the same length and the same direction, even if they start at different places.
• Negative of a vector: same length, opposite direction. BA = –AB.

Two vectors of the same length are not necessarily equal or collinear: they may point in completely different directions.

A vector can be moved anywhere without changing it, as long as its length and direction stay the same. That is why such vectors are called free vectors.