Combining & Resolving Vectors (SL IB Physics)

• Vectors can be changed in a variety of ways, such as
  • Combining through vector addition or subtraction
  • Combining through vector multiplication
  • Resolving into components through trigonometry

Combining Vectors

• Vectors can be combined by adding or subtracting them to produce the resultant vector
  • The resultant vector is sometimes known as the ‘net’ vector (e.g. the net force)

• There are two methods that can be used to combine vectors: the triangle method and the parallelogram method

Triangle 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

Parallelogram method

• 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

Vector Multiplication

• The product of a scalar and a vector is always a vector

• For example, consider the scalar quantity mass  and the vector quantity acceleration 
• The product of mass  and acceleration  gives rise to a vector quantity force 

• For another example, consider the scalar quantity mass  and the vector quantity velocity 
• The product of mass  and velocity  gives rise to a vector quantity momentum 

Resolving Vectors

• Two vectors can be represented by a single resultant vector
  • Resolving a vector is the opposite of adding vectors

• A single resultant vector can be resolved
  • This means it can be represented by two vectors, which in combination have the same effect as the original one

The magnitude of the resultant vector is found by using Pythagoras’ Theorem

• When a single resultant vector is broken down into its parts, those parts are called components
• For example, a force vector of magnitude F_R and an angle of θ to the horizontal is shown below

Resolving two force vectors F₁ and F₂ into a resultant force vector F_R

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

The resultant force F_R can be split into its horizontal and vertical components

• The direction of the resultant vector is found from the angle it makes with the horizontal or vertical
  • The question should imply which angle it is referring to (i.e. calculate the angle from the x-axis)

• Calculating the angle of this resultant vector from the horizontal or vertical can be done using trigonometry
  • Either the sine, cosine or tangent formula can be used depending on which vector magnitudes are calculated

• For the horizontal component, F_x = F cos θ
• For the vertical component, F_y = F sin θ

• In physics, vectors appear in many different topic areas
  • Specifically, vectors are often combined and resolved to solve problems when considering motion, forces, and momentum

• Forces vector diagrams are often represented by free-body force diagrams
• The rules for drawing a free-body diagram are the following:
  • Rule 1: Draw a point in the centre of mass of the body
  • Rule 2: Draw the body free from contact with any other object
  • Rule 3: Draw the forces acting on that body using vectors with length in proportion to its magnitude
  • Rule 4: Draw the tail of the vector from the centre of mass and use the tip to indicate the direction

Point particle representation of the forces acting on a moving object on a rough horizontal surface

• The below example shows the forces acting on an object suspended from a stationary rope

Free-body diagram of an object suspended from a stationary rope

Forces on an Inclined Plane

• A common scenario is an object on an inclined plane
• An inclined plane, or a slope, is a flat surface tilted at an angle, θ

The weight vector of an object on an inclined plane can be split into its components parallel and perpendicular to the slope

• Inclined slope problems can be simplified by considering the components of the forces as parallel or perpendicular to the slope

• The weight (W = mg) of the object is always directed vertically downwards  
• On the inclined slope, weight can be split into the following components:

Perpendicular to the slope: 

Parallel to the slope:  

• The normal (or reaction) force R is always vertically upwards, or perpendicular to the surface

• If there is no friction, the parallel component of weight, mg sin θ, causes the object to move down the slope
• If the object is not moving perpendicular to the slope, the normal force is R = mg cos θ

Equilibrium

• Coplanar forces can be represented by vector triangles
• Forces are in equilibrium if an object is either
  • At rest
  • Moving at constant velocity

• In equilibrium, coplanar forces are represented by closed vector triangles
  • The vectors, when joined together, form a closed path

• The most common forces on objects are
  • Weight
  • Normal reaction force
  • Tension (from cords and strings)
  • Friction
• The forces on a body in equilibrium are demonstrated below:

Three forces on an object in equilibrium form a closed vector triangle