Worked examples: QCE Specialist Mathematics Formula Book

Selected worked examples

1. Vectors · dot product

   Formula or reference: u · v = |u||v|cos θ. In coordinates multiply matching components and add.

   Units and conditions: Use θ between the two nonzero vectors.

   Worked example: For u = (2,1,2) and v = (1,2,−2), u · v = 0, so the vectors are perpendicular.

2. Complex numbers · powers in polar form

   Formula or reference: Raise the modulus to the given integer power and multiply the argument by that power.

   Units and conditions: Use one angle convention throughout.

   Worked example: A number with modulus 2 and argument π/6 has cube 8i.