Topic 6 of 8 · Form 6
Pure 3: vectors and complex numbers
Position vectors, magnitude, unit vectors; scalar product and angles; vector equations of lines r = a + tb; intersection, parallel and skew lines; perpendicular distance from a point to a line.
Notes for this topic 1 file, in order
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Vectors in 3D
Position vectors, magnitude, unit vectors; scalar product and angles; vector equations of lines r = a + tb; intersection, parallel and skew lines; perpendicular distance from a point to a line.
Worked example: Find the angle between i + 2j + 2k and 2i − j + 2k.
Dot product = 2 − 2 + 4 = 4; magnitudes 3 and 3 → cos θ = 4/9 → θ = 63.6°
Complex numbers
Arithmetic in Cartesian form; conjugates; the Argand diagram; modulus and argument; polar and exponential forms re^(iθ); multiplication and division in polar form; square roots; roots of polynomial equations (conjugate pairs); loci (|z − a| = r, |z − a| = |z − b|, arg(z − a) = θ).
Worked example: Find the square roots of 5 + 12i.
(a + ib)² = 5 + 12i → a² − b² = 5, ab = 6 → a = 3, b = 2 → ±(3 + 2i)
Check yourself
- Find |3 − 4i| and arg(−1 + i).
- Write 2e^(iπ/3) in the form x + iy.
- What locus does |z − 2| = |z + 2i| represent?
Show answers
- 5; 3π/4
- 1 + i√3
- The perpendicular bisector of the points 2 and −2i.
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