• 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