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Geometry

Vectors

Vectors build in a straight line: notation, then arithmetic, then finding a route through a diagram, then ratios, and finally collinearity proofs. Each step below assumes the one before it, so work down rather than jumping to the proofs.

Foundation and HigherAQA · Edexcel · OCR · Eduqas

The five steps, in order

  1. 1

    Vector notation

    A vector has both magnitude (size) and direction. Written as a column vector, the top number is movement right and the bottom number is movement up.

    Examplea = (3, −2) means 3 right and 2 down.

  2. 2

    Adding, subtracting and scaling

    Add or subtract vectors by combining their components separately. Multiplying by a scalar multiplies both components.

    Example(2, 3) + (1, −1) = (3, 2), and 3 × (2, 1) = (6, 3).

  3. 3

    Route-finding

    To travel from A to B, find any path through known vectors, usually via the origin. Reversing a vector makes it negative.

    ExampleAB = AO + OB = −a + b, which is written b − a.

  4. 4

    Midpoints and ratios

    For a midpoint, start at one end and travel half the vector. For a ratio m:n, travel m/(m+n) of the journey.

    ExampleIf M is the midpoint of AB, then OM = a + ½(b − a) = ½(a + b).

  5. 5

    Collinearity proofs

    Three points are collinear if one vector between them is a scalar multiple of another and the two share a point. You must state both parts.

    ExampleIf AC = 3AB, then AC is parallel to AB and they share point A, so A, B and C lie on a straight line.

Where marks get lost

  • Write the conclusion out in words. A collinearity proof only gets full marks if you say the vectors are parallel and that they share a common point. The algebra alone is not the proof.
  • Keep vectors in terms of a and b for as long as you can. Substituting numbers early is where most route-finding marks disappear.
  • Watch the direction of travel. BA is the negative of AB, and getting the sign backwards flips the rest of the question.
  • Simplify to a scalar multiple deliberately. Examiners want to see the k in AC = k × AB written down, not implied.

Questions students ask

How do you prove points are collinear with vectors?

Show that one vector is a scalar multiple of another, which makes them parallel, and then point out that the two vectors share a common point. Both halves are needed: parallel vectors that do not share a point are just parallel lines.

What does a column vector actually mean?

The top number is how far you move right and the bottom number is how far you move up. Negatives reverse those directions, so (3, −2) is three right and two down.

Are vectors Foundation or Higher?

Notation, addition, subtraction and scaling appear at both tiers. Route-finding, ratios and collinearity proofs are Higher only, and the proofs are typically among the last questions on the paper.

Why is AB written as b − a?

Because travelling from A to B means going back to the origin first and then out to B: AB = AO + OB. AO is the reverse of OA, so it is −a, giving −a + b, which is normally written b − a.

Reading it is not the same as being able to do it

Every rule above has practice questions behind it that regenerate with new numbers, marked the way the exam marks them.

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