Combining Vectors (OCR A Level Physics)

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 to produce the resultant vector

  • The resultant vector is sometimes known as the ‘net’ vector (eg. the net force)

• There are two methods that can be used to add vectors

  • Calculation – if the vectors are perpendicular

  • Scale drawing – if the vectors are not perpendicular

Vector Calculation

• Vector calculations will be limited to two vectors at right angles

• This means the combined vectors produce a right-angled triangle and the magnitude (length) of the resultant vector is found using Pythagoras’ theorem

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

• 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 (ie. 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

The direction of vectors is found by using trigonometry

Scale Drawing

• When two vectors are not at right angles, the resultant vector can be calculated using a scale drawing

  • Step 1: Link the vectors head-to-tail if they aren’t already

  • Step 2: Draw the resultant vector using the triangle or parallelogram method

  • Step 3: Measure the length of the resultant vector using a ruler

  • Step 4: Measure the angle of the resultant vector (from North if it is a bearing) using a protractor

A scale drawing of two vector additions. The magnitude of resultant vector R is found using a rule and its direction is found using a protractor

• Note that with scale drawings, a scale may be given for the diagram such as 1 cm = 1 km since only limited lengths can be measured using a ruler

• The final answer is always converted back to the units needed in the diagram

  • Eg. For a scale of 1 cm = 2 km, a resultant vector with a length of 5 cm measured on your ruler is actually 10 km in the scenario

• 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

Vector Addition

Vector Subtraction

Combining Vectors with a Vector Triangle

• Coplanar forces can be represented by the resultant of any two 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