Scalars and Vectors

In physics, a physical quantity tells us something we can measure. Some quantities need only a size (how much). Others need a size and a direction (which way). These are called scalars and vectors.

Scalars: size only

A scalar has magnitude (size) only. It does not include direction. Think of the temperature on a thermometer or the mass of a bag. Saying “8 kg” is complete.

• Examples (scalars): distance, speed, time, mass, energy, temperature
• Units (examples): m, s, kg, J, °C

Vectors: size and direction

A vector has magnitude and a direction. Imagine pushing a box: how hard you push and which way both matter.

• Examples (vectors): force, weight, velocity, acceleration, momentum, electric field strength, gravitational field strength
• Representation: arrows. Arrow length shows size; arrow tip shows direction. Vectors may be written like $\vec{v}$ or in bold.

Similar pairs to compare

• Distance (scalar) vs Displacement (vector): distance is “how far you travelled”; displacement is “your straight-line change in position and direction from start to finish”.
• Speed (scalar) vs Velocity (vector): speed is “how fast”; velocity is “how fast and in which direction”.

Combining vectors (resultant)

To combine vectors, place them head-to-tail and draw the single arrow from start to finish. For two vectors at right angles (like east then north), you can use Pythagoras and trigonometry.

Magnitude of resultant $R$ when components are $A$ and $B$:

$$R = \sqrt{A^2 + B^2}$$

Direction angle $\theta$ (from $A$ towards $B$):

$$\theta = \tan^{-1}\!\left(\dfrac{B}{A}\right)$$