# 1.1.4 Scalars & Vectors

### What are Scalar & Vector Quantities?

• A scalar is a quantity which only has a magnitude (size)
• A vector is a quantity which has both a magnitude and a direction
• For example, if a person goes on a hike in the woods to a location which is a couple of miles from their starting point
• As the crow flies, their displacement will only be a few miles but the distance they walked will be much longer

Displacement is a vector while distance is a scalar quantity

• Distance is a scalar quantity because it describes how an object has travelled overall, but not the direction it has travelled in
• Displacement is a vector quantity because it describes how far an object is from where it started and in what direction
• There are a number of common scalar and vector quantities

Scalars and Vectors Table

#### Exam Tip

Do you have trouble figuring out if a quantity is a vector or a scalar? Just think – can this quantity have a minus sign? For example – can you have negative energy? No. Can you have negative displacement? Yes!

### Combining Vectors

• Vectors are represented by an arrow
• The arrowhead indicates the direction of the vector
• The length of the arrow represents the magnitude
• Vectors can be combined by adding or subtracting them from each other
• There are two methods that can be used to combine vectors: the triangle method and the parallelogram method
• To combine vectors using the triangle method:
• Step 2: the resultant vector is formed by connecting the tail of the first vector to the head of the second vector
• To combine vectors using the parallelogram method:
• Step 1: link the vectors tail-to-tail
• Step 2: complete the resulting parallelogram
• Step 3: the resultant vector is the diagonal of the parallelogram
• When two or more vectors are added together (or one is subtracted from the other), a single vector is formed and is known as the resultant vector

#### Condition for Equilibrium

• Coplanar forces can be represented by vector triangles
• In equilibrium, these are closed vector triangles. The vectors, when joined together, form a closed path

If three forces acting on an object are in equilibrium; they form a closed triangle

### Resolving Vectors

• Two vectors can be represented by a single resultant vector that has the same effect
• A single resultant vector can be resolved and represented by two vectors, which in combination have the same effect as the original one
• When a single resultant vector is broken down into its parts, those parts are called components
• For example, a force vector of magnitude F and an angle of θ to the horizontal is shown below

• It is possible to resolve this vector into its horizontal and vertical components using trigonometry

• For the horizontal component, Fx = Fcosθ
• For the vertical component, Fy = Fsinθ

### Author: Ashika

Ashika graduated with a first-class Physics degree from Manchester University and, having worked as a software engineer, focused on Physics education, creating engaging content to help students across all levels. Now an experienced GCSE and A Level Physics and Maths tutor, Ashika helps to grow and improve our Physics resources.
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