3.7.6 Components of Vectors

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 θ
• This is equivalent to 
• The component of a acting perpendicular to b will be equal to the product of the magnitude of a and the sine of the angle θ
• The component of a acting perpendicular to the direction of b equals |a|sin θ
• This is equivalent to  
• The 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