Plane Vectors and Complex Numbers
Plane vectors and complex numbers
A vector has magnitude and direction; in coordinates, addition and scalar multiplication are componentwise, while the dot product a·b=|a||b|cos θ measures projection and detects perpendicularity. Know collinearity (b=λa), the projection formula, and the geometric meaning of the dot product's sign. For complex numbers, use z=a+bi, conjugation, modulus, and argument; multiplication adds arguments and multiplies moduli. Division is done by multiplying numerator and denominator by the conjugate.
Traps: confusing a·b=0 (perpendicular) with a×b=0 in space (parallel); forgetting the sign of a projection; writing |z|² as z² instead of z z̄; and placing an argument in the wrong quadrant. Also remember that two complex numbers are equal only when both real and imaginary parts match, and that i²=−1.
Worked example: z=(1+i)/(1−i). Multiply by (1+i)/(1+i): numerator (1+i)²=2i, denominator 1+1=2, so z=i. Hence |z|=1 and arg z=π/2. For vectors a=(1,2) and b=(2,−1), a·b=2−2=0, so they are perpendicular. Rule of thumb: choose coordinates for vector geometry and conjugate multiplication for complex division; then check both modulus and quadrant.