Maths Essentials, Unit 27, Lesson 3 of 4

Vectors in shapes

GCSE Maths, Vectors, 25 minutes

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Writing vectors in triangles, parallelograms and hexagons in terms of a and b, including midpoints and ratios.

What you will learn

  • Find a route through a shape and write it in terms of a and b
  • Use midpoints and ratios on a line
  • Use parallel and equal sides of shapes

Find a route

To write AB in terms of <b>a</b> and <b>b</b>, find a route from A to B along vectors you know.

Going with an arrow adds it; going against an arrow subtracts it.

abOAB

Not drawn accurately

  1. AB = AO + OB
  2. AB = −a + b

Any route works, as long as each step is one you know. Go round step by step, with a sign for each step: in OABC with OA = a, AB = b and BC = 2a − b, OC = a + b + (2a − b) = 3a. Collect the a terms and the b terms to simplify.

In a parallelogram OABC, opposite sides are equal vectors: if OA = a and OC = b, then CB = a and AB = b, so AC = −a + b. A regular hexagon with centre O is six equilateral triangles: each side is equal and parallel to a line from O, so if OA = a and OB = b, then BC = −a (it is parallel to AO) and CD = −b.

When a line is extended, use the multiple given: if OC = 3a on the line OA, then BC = BO + OC = −b + 3a.

The rest of this lesson and its 8 practice questions are locked.

In this lesson

  1. Find a route
  2. Midpoints and ratios
  3. A vector from two others (Higher)
  4. Try it: First question

8 practice questions, marked as you go; about 25 minutes in all.

In this unit: Vectors

Column vectors and how to add, subtract and multiply them, vectors between points, and vector geometry and proof in terms of a and b.

All of Vectors

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