MathematicsExam code: 0580

Vectors

Vectors and Transformations

PractiseFlashcards

Position vectors

  • A position vector gives the location of a point relative to the origin, so the position vector of A is OA
  • A question asking for "the position vector of P" wants OP, so your route must start at O
  • The vector between two points is found by going backwards to the origin and forwards again: AB = OB − OA
  • Reading that as "end minus start" is the quickest way to remember it
Three points A(−4, 2), B(2, 4) and C(3, −4) plotted on axes and joined by arrows labelled AB, AC and CB, pointing from A to B, from A to C and from C to B
Source: Vector diagrams by Save My Exams
A triangle OAB with the position vector of A written as OA = a and the position vector of B as OB = b, and the vector AB worked out by travelling back to O and out again: AB = −OA + OB = b − a
Source: Position & displacement vectors by Save My Exams

Worked example

From coordinates to a column vector and back

A is the point (−3, 4) and B is the point (2, −1). Find the vector BA. Then find the coordinates of C, where BC = (−7, 3) written as a column vector.

Solution:

  • BA goes from B to A, so take B's coordinates away from A's: end minus start
  • Top: −3 − 2 = −5
  • Bottom: 4 − (−1) = 5
  • So BA = (−5, 5) as a column vector, 5 left and 5 up
  • For C, start at B and add the movement BC
  • x: 2 + (−7) = −5, and y: −1 + 3 = 2
  • So C is the point (−5, 2), written as coordinates because the question asks for coordinates

Exam tip

The vector BA starts at B

A vector written the wrong way round, such as AB given when BA was asked for, has both signs flipped, and the schemes give it only one of the two marks, no more than an answer with just one number right. When a question names A and then B but asks for BA, sketch the move from B to A before you subtract, and check that your signs match it.

Finding a path

  • Any vector across a diagram can be built by travelling along vectors you already know, in any route that works
  • Travelling with an arrow adds that vector; travelling against it subtracts
  • Different routes give different-looking expressions that simplify to the same thing, and mark schemes accept any correct path
A grid of identical parallelograms with vertices labelled A to T, where AB is the vector a and AF is the vector b: the route from F to B is traced in red as going back along b and then along a, giving FB = −b + a
Source: Finding vector paths by Save My Exams
  • Collect like terms at the end, because "give your answer in its simplest form" appears on almost every one of these questions

Exam tip

Write the route before you simplify

Questions asking for a vector in terms of two given vectors are set in 11 of the 21 papers. Write the route in letters first, such as OM = OA + AM: the schemes give a mark for a correct route along the lines of the diagram even if nothing else is right. A correct but unsimplified answer usually earns every mark except the last, which needs the like terms collected.

Worked example

A point dividing a line in a ratio

In triangle OAB, OA = a and OB = b. The point P lies on AB so that AP : PB = 2 : 3. Find OP in terms of a and b, simplified as far as possible.

Triangle OAB with O at the bottom left. An arrow from O to A is labelled a and an arrow from O to B is labelled b. The point P is marked on the line AB, nearer to A than to B. The figure is marked NOT TO SCALE.

Solution:

  • Travel from O to P by going along OA and then part of AB
  • First find the whole of AB: AB = OB − OA = b − a
  • The ratio 2 : 3 splits AB into 5 equal parts, and P is 2 of them along from A
  • So AP = ⅖(b − a)
  • Now OP = OA + AP = a + ⅖(b − a)
  • Expand: OP = a + ⅖b − ⅖a
  • Collect the a terms: OP = ⅗a + ⅖b
  • So OP = ⅗a + ⅖b
  • Check: at the far end, O to B would give a + 1(b − a) = b, as it should

Build on this topic

Check your understanding17 exam questions, marked instantlyTest your recall26 flashcards on the key facts