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Topic 9 of 11 · Form 3

Vectors and transformations

Column vectors; addition, subtraction and multiplication by a scalar; magnitude |(x, y)| = √(x² + y²); position vectors; vector geometry (showing points are collinear or lines parallel).

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Vectors

Column vectors; addition, subtraction and multiplication by a scalar; magnitude |(x, y)| = √(x² + y²); position vectors; vector geometry (showing points are collinear or lines parallel).

Worked example: a = (3, −4). Find |a| and 2a.

|a| = 5; 2a = (6, −8)

Worked example: OA = a, OB = b, and M is the midpoint of AB. Express OM in terms of a and b.

AB = b − a → OM = a + ½(b − a) = ½(a + b)

Transformations

  • Reflection (state the mirror line), rotation (centre, angle, direction), translation (vector), enlargement (centre and scale factor, including fractional and negative factors).
  • Describe a single transformation fully.
  • Combined transformations.

Enlargement

Lengths multiply by the scale factor k; areas by k². A negative scale factor inverts the image through the centre.

Exam tip: For a full description of a rotation, give the centre, the angle and the direction – missing any one loses marks.

Check yourself

  1. Find the magnitude of (5, 12).
  2. Describe the transformation mapping (2, 3) to (3, 2) for every point.
  3. A shape of area 6 cm² is enlarged by scale factor 3. Find the new area.
Show answers
  1. 13
  2. Reflection in the line y = x.
  3. 54 cm²
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