Specialist Mathematics · Unit 3 · Vectors in two and three dimensions · Vector and Cartesian equations
Determine 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 𝑑3
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Define 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)
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Understand and use equations of spheres. equation of sphere: (𝑥 − ℎ)2 + (𝑦 − 𝑘)2 + (𝑧 − 𝑙)2 = 𝑟2