Scalars and vectors (AQA A-Level Physics): Revision Notes

4.1.1 Scalars and vectors

In physics, quantities can be classified as scalars or vectors:

• Scalars have only magnitude (size) and no direction. Examples include distance, speed, mass, and temperature.
• Vectors have both magnitude and direction. Examples include displacement, velocity, force (or weight), and acceleration.

Adding Vectors

There are two main methods to add vectors, depending on their orientation:

1. Calculation Method (for Perpendicular Vectors):

• When vectors are perpendicular to each other, you can use Pythagoras' theorem to find the resultant vector's magnitude and trigonometry to determine its direction.

2. Scale Drawing (for Non-Perpendicular Vectors):

• For vectors at arbitrary angles, use a scale diagram. Draw each vector to scale, using a ruler and protractor, then measure the resultant vector's magnitude and direction.

Resolving Vectors

The process of breaking a vector into its horizontal and vertical components is called resolving. This is useful for analysing forces in different directions.

For a vector $V$ at an angle $\theta$ :

• Horizontal Component: $x = V \cos \theta$
• Vertical Component: $y = V \sin \theta$ Tip: If moving through the angle to reach the component, use cos. If moving away from the angle, use sin.

Equilibrium and Vector Triangles

An object is in equilibrium if the sum of all forces acting on it is zero, meaning it has no resultant force and remains at rest or in uniform motion (Newton's First Law).

To show equilibrium:

• Add horizontal and vertical components of forces to check if they balance.
• For three forces, draw a scale triangle: if it forms a closed triangle, the object is in equilibrium.