Combining & Resolving Vectors (DP IB Physics): Revision Note

Combining & resolving vectors

• 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

  • To subtract vectors, change the direction of the vector from positive to negative and add them in the same way

Triangle method for adding and subtracting vectors

The triangle method links vectors tip to tail to find the resultant 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

Parallelogram method for adding and subtracting vectors

The parallelogram method links vectors tail to tail to find the resultant vector 

• When two or more vectors are added together (or one is subtracted from the other), a single vector is formed, known as the resultant vector

• The magnitude of the resultant vector can be found using Pythagoras' theorem or trigonometry

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