Mr Kidman's Classroom← Back

Edexcel · GCSE Maths · 1MA1 · Foundation and Higher

M35 · Vectors

PLCWordPLCPDFMind map

Revision notes, worked examples and methods for vectors.

Notes and quizzes ready · 50 questions · Revision video ready.

Revise the key ideas

Vector notation and arithmetic

  • A vector has magnitude and direction. A column vector (4−3) represents 4 units right and 3 down. Its components are signed displacements, not a coordinate pair identifying one fixed point.
  • Equal vectors have equal components even when drawn in different positions. The negative vector reverses direction: −(4−3) = (−43).
  • Add component by component: (23) + (4−1) = (62). Geometrically, place the second arrow's tail at the first arrow's head.
    Head-to-tail vector additionA vector two right and three up is followed by four right and one down. The resultant is six right and two up.a = (2, 3)b = (4, −1)a + b = (6, 2)Place the second vector head-to-tail
    Head-to-tail vector addition
    Enlarge diagram: Head-to-tail vector addition
  • Subtract using the reverse vector: (52) − (14) = (4−2). Order matters.
  • Multiplying by a scalar changes length and may reverse direction. 3(2−1) = (6−3), while −2(2−1) = (−42).
  • If A = (1, 2) and B = (5, −1), vector AB is B − A = (4−3). Vector BA is its negative.
  • A vector's magnitude follows Pythagoras: |(34)| = √(32 + 42) = 5. A translation's vector is measured in the coordinate system's units.

Higher — vector geometry and proofs

  • Use bold letters such as a or directed labels such as AB→ for vectors. If OA = a and OB = b, then AB = b − a.
  • The midpoint M of AB has OM = (a + b). If M divides AB in ratio 1 : 2 from A, OM = a + (b − a).
  • Vectors that are non-zero scalar multiples are parallel. To prove three points are collinear, show vectors along two joining segments are scalar multiples and share a point.
  • Worked example: In triangle OAB, midpoints M of OA and N of OB have OM = a and ON = b. Therefore MN = (b − a) = AB, proving MN is parallel to AB and half its length.
    Midpoint vector routes. Triangle OAB with midpoint and labelled vector routes
  • For a parallelogram OABC in order, if OA = a and OC = b, then OB = a + b. Diagonals can be expressed in more than one route to locate their intersection.
  • Direction matters in each route: AB + BC = AC, but AB + CB is different. Draw arrows and keep start/end labels consistent.
  • For an intersection problem, express its position along each line with parameters, equate coefficients of independent vectors and solve the resulting simultaneous equations.

Test yourself

50 questions · Sets of 10 from the selected tier. For fractions, use / when typing; for powers, use superscripts or ^. Follow each question's answer format. These quick checks support revision; practise full written solutions and proofs too.

Watch M35 · Vectors

Revise vectors with this narrated video. Use the player controls to pause, seek, adjust the volume or mute. Turn English captions on or off using the captions menu.

Open or download the video · English captions

Mind map

Use the branches to recall the ideas and explain their connections. Check the revision notes for the full detail.

View M35 mind map
M35 M35 mind map: Vectors, Arithmetic, Positions, Proof / ratios. A text version follows.
Open the full-size map to zoom. Download the PDF to print on A4 or enlarge to A3.

Open full-size map Download A4 PDF

Read the mind map as text

Vectors

  • Components: Magnitude and direction; equal components, different positions
  • Length: Magnitude √(x²+y²); (3,4) has length 5

Arithmetic

  • Add / subtract: Component by component; head-to-tail; subtraction reverses
  • Scalar: Scales length; negative reverses direction

Positions

  • Direction: Higher: AB=B−A; if OA=a,OB=b then AB=b−a
  • Routes: Higher: AB+BC=AC; keep arrow start/end labels consistent

Proof / ratios

  • Divide: Higher: Midpoint (a+b)/2; ratio1:2 from A → a+(b−a)/3
  • Parallel / collinear: Higher: Non-zero scalar multiples; collinear shares common point
  • Intersection: Higher: Use two routes; equate independent vector coefficients

Connections

  • Arithmetic → Positions: Component arithmetic agrees with directed routes
  • Positions → Proof / ratios: Scalar multiples prove parallel relationships