Vectors and Scalars

In short: A scalar is a quantity with size only, like distance, speed, mass or time. A vector has size and direction, like displacement, velocity, acceleration or force, and is drawn as an arrow.

Scalars and vectors

A scalar has size only (also called magnitude). A vector has size and direction.

Drawing vectors

We draw a vector as an arrow.

• The length of the arrow shows the size.
• The arrowhead shows the direction.

Adding vectors in a straight line

When vectors point along the same line, choose one direction as positive. The opposite direction is negative. Then just add.

• 5 m east + 3 m east = 8 m east
• 5 m east + 3 m west = 5 + (−3) = 2 m east

The answer you get when you add vectors is called the resultant.

Distance and displacement

• Distance = total length of the path. A scalar.
• Displacement = straight line from start to finish, with direction. A vector.

Zara walks 4 m east, then 3 m north. Her distance is $4+3=7$ m. Her displacement is the straight line from start to end. Using Pythagoras:

$$s=\sqrt{4^2+3^2}=\sqrt{25}=5\text{ m}$$

So she ends up 5 m from the start, in a north-east direction.

Speed and velocity

• Speed = distance ÷ time (scalar)
• Velocity = displacement ÷ time (vector)

A car going round a roundabout at a steady 30 km/h has constant speed but changing velocity, because its direction keeps changing.

Quick summary

• Scalars have size only. Vectors have size and direction.
• Distance, speed, mass, time, energy are scalars.
• Displacement, velocity, acceleration, force are vectors.
• Draw vectors as arrows. Length shows size, the head shows direction.
• In a straight line, pick a positive direction and add with signs. The answer is the resultant.
• At right angles, use Pythagoras for the size.

Quiz