Why do we write vectors in component form?

• When working with vectors in context it is often useful to break them down into components acting in a direction that is not one of the base vectors
• The?base vectors?are vectors acting in the directions?i,?j?and?k
• The vector will need to be resolved into components that are acting?perpendicular?to each other
• Usually, one component will be acting?parallel?to the direction of another vector and the other will act?perpendicular?to the direction of the vector
• For example: the components of a?force?parallel and perpendicular to the?line of motion?allows different types of problems to be solved
  • The?parallel?component of a force acting directly on a particle will be the component that causes an effect on the particle
  • The?perpendicular?component of a force acting directly on a particle will be the component that has no effect on the particle
• The two components of the force will have the same combined effect as the original vector

How do we write vectors in component form?

• Use?trigonometry?to resolve a vector acting at an angle
• Given a vector?a?acting at an angle θ to another vector?b
  • Draw a vector triangle by decomposing the vector?a?into its components parallel and perpendicular to the direction of the vector?b
• The vector?a?will be the?hypotenuse?of the triangle and the two components will make up the?opposite?and?adjacent?sides
• The component of?a?acting?parallel?to?b?will be equal to the product of the magnitude of?a?and the cosine of the angle θ
  • The component of?a?acting in the direction of?b?equals |a|cos θThe formulae for the components using the?scalar product?and the?vector product?are particularly useful as the angle is not needed
• The question may give you the angle the vector is acting in as a bearing
  • Bearings are always the angle taken from the north