amazon.co.ukVectors
A-Level Further Mathematics revision: Vectors. Learning objectives, key points, worked examples and practice questions across AQA, Edexcel, OCR, WJEC and CCEA.
📌 Key Points
Key Fact: Vector form: r = a + λb (line), r = a + λb + μc (plane)
Key Fact: Dot product: a·b = |a||b|cosθ = a₁b₁+a₂b₂+a₃b₃; perpendicular iff a·b = 0
Key Fact: Cross product: axb = |a||b|sinθ n̂ = det[i j k; a₁ a₂ a₃; b₁ b₂ b₃]
Key Fact: Area: triangle = ½|axb|; parallelogram = |axb|; volume = |a·(bxc)|
Key Fact: Line eqn: r = a + λb; Cartesian: (x-x₁)/l = (y-y₁)/m = (z-z₁)/n
Key Fact: Plane eqn: r·n = d (normal n, distance d/|n| from origin) or n₁x+n₂y+n₃z = d
Key Fact: Intersection: substitute line into plane, solve for λ
Key Fact: Distance point to plane: |ax₀+by₀+cz₀-d|/sqrt(a^2+b^2+c^2); point to line: |bx(a-p)|/|b|
🎯 Learning Objectives
- Perform vector operations in 2D and 3D
- Use scalar (dot) product to find angles and test perpendicularity
- Use vector (cross) product to find area and normal vectors
- Find equations of lines in vector and cartesian form
- Find equations of planes in vector, cartesian and parametric form
- Calculate distances: point-to-line, point-to-plane, line-to-line, line-to-plane
- Find intersections of lines and planes
- Apply vectors to mechanics problems
💡 Worked Example
Exam-Style Question
Question: Find the intersection of line r = [1; 2; 3] + λ[2; -1; 4] with plane 2x - y + 3z = 10
Model Answer:
Substitute: x=1+2λ, y=2-λ, z=3+4λ. 2(1+2λ)-(2-λ)+3(3+4λ)=10 -> 2+4λ-2+λ+9+12λ=10 -> 17λ+9=10 -> λ=1/17. Point: [19/17; 33/17; 55/17]
❓ Practice Questions
Questions:
- Find angle between lines r = [1; 0; -1] + λ[2; 1; 3] and r = [0; 1; 1] + μ[1; -2; 2]
- Find equation of plane through (1,2,3) perpendicular to vector [2; -1; 1]
- Distance from point (1,2,3) to plane 2x - 3y + 6z = 10
- Find shortest distance between skew lines r = [1; 0; 0] + λ[1; 1; 0] and r = [0; 1; 1] + μ[0; 1; 1]
- Volume of tetrahedron with vertices (0,0,0), (1,0,0), (0,2,0), (0,0,3)
🎬 Video Resources
- Vectors - Overview -- YouTube Search
- Vectors - Exam Questions -- YouTube Search
📄 Past Papers & Exam Resources
🔗 Further Reading & Resources
- Physics & Maths Tutor - Vectors -- Free notes, worksheets, and past paper questions by topic
- Dr Frost Maths - Vectors -- Free resources, slides, and interactive questions
- BBC Bitesize - Vectors -- BBC Bitesize revision resources
📚 Lesson Plan (50 minutes)
- Starter (5 min): Recall prior knowledge of vectors with quick questions.
- Teaching (15 min): Work through each of the learning objectives, explaining principles step by step.
- Key points review (5 min): Revisit the key points together, confirming understanding.
- Worked example (10 min): Model the example question: Find the intersection of line r = [1; 2; 3] + λ[2; -1; 4] with plane 2x - y + 3z = 10. Solution: Substitute: x=1+2λ, y=2-λ, z=3+4λ. 2(1+2λ)-(2-λ)+3(3+4λ)=10 -> 2+4λ-2+λ+9+12λ=10 -> 17λ+9=10 -> λ=1/17. Point: [19/17; 33/17; 55/17]
- Practice (10 min): Students attempt the practice questions independently; circulate and support.
- Plenary (5 min): Review answers and address misconceptions.
🏠 Homework
- Find angle between lines r = [1; 0; -1] + λ[2; 1; 3] and r = [0; 1; 1] + μ[1; -2; 2]
- Find equation of plane through (1,2,3) perpendicular to vector [2; -1; 1]
- Distance from point (1,2,3) to plane 2x - 3y + 6z = 10
- Find shortest distance between skew lines r = [1; 0; 0] + λ[1; 1; 0] and r = [0; 1; 1] + μ[0; 1; 1]
- Volume of tetrahedron with vertices (0,0,0), (1,0,0), (0,2,0), (0,0,3)
🧾 Assessment
Check practice answers against the model answer; use the built-in practice questions as formative assessment.