1.1.4 Scalars & Vectors

• 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

• 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 1: link the vectors head-to-tail
  • 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

Vector Addition

Vector Subtraction

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

• 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, F_x = Fcosθ
• For the vertical component, F_y = Fsinθ