Vector and Cartesian equations
Specialist Mathematics · Unit 3 — Further complex numbers, proof, vectors and matrices · Vectors in two and three dimensions
Learning objectives (5)
LO-1Define and use the vector (cross) product to determine a vector normal to a given plane, with and without technology. 𝒂 × 𝒃 = |𝒂| |𝒃| sin(𝜃) 𝒏̂ 𝒂 × 𝒃 = ( 𝑎1 𝑎2 𝑎3) × ( 𝑏1 𝑏2 𝑏3) = ( 𝑎2𝑏3 − 𝑎3𝑏2 𝑎3𝑏1 − 𝑎1𝑏3 𝑎1𝑏2 − 𝑎2𝑏1)LO-2Determine vector, parametric and Cartesian equations of straight lines and straight-line segments given the position of two points, or equivalent information, in both two and three dimensions. vector equation of line: 𝒓 = 𝒂 + 𝑡𝒅 parametric equations of line: 𝑥 = 𝑎1 + 𝑡 𝑑1 𝑦 = 𝑎2 + 𝑡 𝑑2 𝑧 = 𝑎3 + 𝑡 𝑑3 Cartesian equation of line: 𝑥−𝑎1 𝑑1 = 𝑦−𝑎2 𝑑2 = 𝑧−𝑎3 𝑑3LO-3Understand and use equations of spheres. equation of sphere: (𝑥 − ℎ)2 + (𝑦 − 𝑘)2 + (𝑧 − 𝑙)2 = 𝑟2LO-4Use vector equations of curves in two or three dimensions involving a parameter, and determine a ‘corresponding’ Cartesian equation in the two-dimensional case.LO-5Use vector methods in applications, including areas of shapes and determining vector and Cartesian equations of a plane and of regions in a plane. vector equation of plane: 𝒓 ⋅ 𝒏 = 𝒂 ⋅ 𝒏 Cartesian equation of plane: 𝑎𝑥 + 𝑏𝑦 + 𝑐𝑧 + 𝑑 = 0
Practise these objectives with instant AI marking
Adaptive questions tied to QCAA mark schemes. Free to start.
Start free practice