Vectors — A-Level Mathematics Revision
Revise Vectors for A-Level Mathematics. Step-by-step explanation, worked examples, common mistakes and exam-style practice aligned to AQA, Edexcel and OCR.
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- Vectors in A-Level Mathematics: explanation, examples, and practice links on this page.
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Go to ProofWhat is Vectors?
Vectors at A-Level are used to represent quantities that have both magnitude and direction. You will learn to perform vector arithmetic, find the magnitude of a vector, and use vectors to solve problems in geometry and mechanics. This includes working with vectors in 2D and 3D.
Board notes: All A-Level Maths boards (AQA, Edexcel, OCR) cover vectors in both pure maths and mechanics. The applications in mechanics, such as resolving forces, are a key part of the applied content.
Step-by-step explanationWorked example
Find the angle between the vectors a = 2i + 3j - k and b = i - 2j + 4k. First, find the scalar product a.b = (2)(1) + (3)(-2) + (-1)(4) = 2 - 6 - 4 = -8. Next, find the magnitudes: |a| = √(2² + 3² + (-1)²) = √14 and |b| = √(1² + (-2)² + 4²) = √21. Now use the formula cos(θ) = a.b / (|a||b|) = -8 / (√14 * √21). So, θ = arccos(-8 / √294) ≈ 117.8°.
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Common mistakes
- 1Confusing position vectors and direction vectors. A position vector gives the location of a point relative to the origin, while a direction vector represents the direction and magnitude of a displacement.
- 2Making errors in vector addition and subtraction. Remember to add or subtract the corresponding components of the vectors.
- 3Incorrectly calculating the scalar product (dot product) of two vectors. The formula is a.b = |a||b|cos(θ), or in component form, a.b = a₁b₁ + a₂b₂ + a₃b₃.
Vectors exam questions
Exam-style questions for Vectors with mark-scheme style solutions and timing practice. Aligned to AQA, Edexcel and OCR specifications.
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Step-by-step method
Step-by-step explanation
4 steps · Worked method for Vectors
Core concept
Vectors at A-Level are used to represent quantities that have both magnitude and direction. You will learn to perform vector arithmetic, find the magnitude of a vector, and use vectors to solve proble…
Frequently asked questions
What is a unit vector?
A unit vector is a vector with a magnitude of 1. To find the unit vector in the direction of a vector a, you divide the vector by its magnitude: a-hat = a / |a|.
How are vectors used in mechanics?
In mechanics, vectors are used to represent quantities like displacement, velocity, acceleration, and force. They allow us to solve problems involving the motion of objects and the forces acting on them.