Vectors?are used throughout mechanics to describe forces and motion

?How are Vectors used in Mechanics?

• Vector?questions are often embedded in a Mechanics context
• Vectors will most commonly represent?forces,?accelerations?or?velocities, they can also represent displacement
• Newton’s Second Law?F?=?ma?is essential
  • Remember that?F?and?a?(force and acceleration) are?vectors, while m (mass) is a?scalar
• When a particle is in?equilibrium, the vector sum of the forces on it is equal to zero

What is vector notation?

• There are?two?vector?notations?used in A level mathematics:
  • Column?vectors: This is one number written above the other enclosed in brackets,

e.g. The (column) vector meaning?3 units?in the?positive?horizontal?(x) direction (i.e.,?right) and?2 units?in the?negative?vertical?(y?) direction (i.e.,?down) can be written as:

• • i?and?j?notation:?i?and?j?are?unit?vectors?(they have?magnitude 1) in the positive horizontal and positive vertical directions respectively

e.g. The vector (-4i?+ 3j) would mean?4 units?in the?negative?horizontal?(x) direction (i.e.?left) and?3 units?in the?positive?vertical?(y) direction (i.e.?up)

• As they are vectors,?i?and?j?are displayed in?bold?in textbooks and online but in handwriting they would be?underlined?(i?and?j)

Calculating Resultant Vectors:

• Both?column vectors?and?i, j notation?can be used for calculating?resultant vectors
• Adding vectors together gives the?resultant vector
• This is the same when adding force vectors; the?resultant force?is simply the force vectors added together
• Forces in?equilibrium?have a resultant force equal to?zero

Calculating Magnitude and Direction:

• Pythagoras is used to find the?magnitude?of a vector.
• Trigonometry is used to find the?direction?of a vector
• You may be asked for the direction as a?bearing.?The?unit vector?j?can be used to represent north and the unit vector?i?can?be used to represent east.
  • In the case where?k?> 0
    • If a particle is moving north then its velocity will have a vector of?kj
    • if a particle?A?is due south of another particle?B?then the displacement vector from?B?to?A?will have the form -kj
    • If a particle is moving east then its velocity will have a vector of?ki
    • if a particle?A?is due west of another particle?B?then the displacement vector from?B?to?A?will have the form -ki
  • If the position vectors of two particles have the same?j?component, then the particle with the greater?i?component will be positioned due east of the other
  • If the position vector of a particle has equal?i?and?j?components then it is positioned due north - east of the origin
    • if a particle?A?is north – east of another particle?B?then the displacement vector from?B?to?A?will have equal?i?and?j?components

Resolving Vectors:

• Resolving?a vector means writing it in?component?form (as?i?and?j?components)
• Given the?magnitude?and?direction?of a vector you can work out its components and vice versa

a)? Find the magnitude of the resultant force?R?acting on the particle.

Leaving your answer as a simplified surd.

b)? Find the bearing of the resultant force?R.

Give your answer to the nearest degree.

A third force?F3?= si?– tj?brings the particle into equilibrium.

c)? ?Find s and t and state the force for?F3?in terms of?i?and?j.