1. For nonzero vectors, a·b=0 implies what?
- They are parallel
- They have equal magnitude
- They are perpendicular
Check the answer and reasoning
They are perpendicular
a·b=|a||b|cos θ, so cos θ=0.
MATH / A2 level
Vectors and complex numbers
Three-dimensional vectors express lines, distances and angles. The scalar product connects components with the angle between nonzero vectors.
Learning objectives
- Explain vectors and complex numbers using a concrete example.
- Apply the worked method and justify each step.
- Recognise the limitation described in the final check.
The big idea
Three-dimensional vectors express lines, distances and angles. The scalar product connects components with the angle between nonzero vectors.
Complex numbers extend real arithmetic using i²=−1. Cartesian and polar forms show different structure; conjugates support division, and roots occupy regular angular positions in an Argand diagram.
Equations and relationships
Product with the complex conjugate.
Worked example
Evaluate (2+3i)(2−3i).
Follow the solution
The cross terms cancel and −9i²=9, giving 4+9=13.
Investigate and explain
Predict the worked result before calculating. Change one input, record the new result, and explain which assumptions still hold.
Related interactive model. Compare its assumptions with the explanation; it is not a complete syllabus lesson.
Check your understanding
2. Evaluate (2+3i)(2−3i).
- 13
- −5
- 4−9i
Check the answer and reasoning
13
The cross terms cancel and −9i²=9, giving 4+9=13.
3. What is the modulus of 3+4i?
- 7
- 5
- 25
Check the answer and reasoning
5
Its distance from the origin is √(3²+4²)=5.