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AQA GCSE Maths

Vectors: notation, arithmetic and geometric proof — AQA GCSE Maths revision

Free revision notes, key terms, common exam traps and 5 practice questions with answers. About 6 minutes to read.

Vector notation and basics

  • A vector has both magnitude (size) and direction, written as column vector (x, y) or bold letter a.
  • Vector AB means the vector from point A to point B; AB = -BA.
  • Magnitude of vector (x, y) is sqrt(x^2 + y^2), found using Pythagoras.

Vector arithmetic

  • To add vectors, add corresponding components; to subtract, subtract components.
  • Scalar multiplication: multiply every component by the scalar, e.g. 3(2,5) = (6,15).
  • Parallel vectors are scalar multiples of each other, e.g. b = 2a means b is parallel to a and twice as long.

Key terms

Vector
A quantity with both magnitude and direction.
Scalar
A quantity with magnitude only, used to multiply vectors and change their size.
Magnitude
The size (length) of a vector, found using Pythagoras.
Position vector
A vector describing the position of a point relative to the origin.
Parallel vectors
Vectors that are scalar multiples of one another.
Collinear
Points that lie on the same straight line.

Common exam traps

  • Forgetting that AB = -BA (direction matters).
  • Adding magnitudes instead of adding vector components.
  • Confusing parallel vectors with equal vectors — parallel just means same or opposite direction.

Practice questions with answers

  1. 1. Add vectors (2,3) and (4,-1).

    Answer: (6,2)

    Add components: (2+4, 3+(-1)) = (6,2).

  2. 2. Find 3 times the vector (2,-5).

    Answer: (6,-15)

    Multiply each component by 3: (3×2, 3×-5) = (6,-15).

  3. 3. Find the magnitude of vector (3,4).

    Answer: 5

    sqrt(3^2+4^2) = sqrt(9+16) = sqrt(25) = 5.

  4. 4. If AB = (5,2), what is BA?

    • • (5,2)
    • • (-5,-2)
    • • (2,5)
    • • (-5,2)

    Answer: (-5,-2)

    BA is the reverse direction of AB, so negate both components.

  5. 5. a = (1,2), b = (3,-1). Find a - b.

    Answer: (-2,3)

    Subtract components: (1-3, 2-(-1)) = (-2,3).

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