A-Level Mathematics Revision — Vectors
Revise Vectors for A-Level Mathematics with a topic explanation, worked example and common mistakes. Check the board notes for specification differences.
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What 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 examples
Worked 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.
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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.