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Cambridge IGCSE0580

Vectors in two dimensions

Mathematics 0580 Chapter Notes

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Vectors in two dimensions
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Vectors in two dimensions notes

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In this chapter

  1. 1Column vectors and vector notation
  2. 2Translation by a vector
  3. 3Adding and subtracting vectors
  4. 4Multiplying a vector by a scalar
  5. 5Vectors in terms of a and b

1. Column vectors and vector notation

A vector is printed as overrightarrowAB, for the vector from A to B, or as a single bold letter such as a. In handwriting, put an arrow above the pair of capital letters or underline the single letter.

A column vector gives the movement in the x-direction on top and in the y-direction below. A negative top number is a move to the left. A negative lower number is a move down.

To find overrightarrowAB, subtract the coordinates of the start from those of the end. Write both components in brackets, with no fraction line.

overrightarrowAB=beginpmatrixx_B-x_A y_B-y_Aendpmatrix

overrightarrowBA=-overrightarrowAB

Examiner insight

“Candidates find it more challenging to write a vector between two points when no diagram is given.” “Fraction lines are penalised and were more noticeable this series and a few just gave one component in their vectors.”

Worked example 12 marks

Point A has coordinates (–4, 3) and point B has coordinates (4, 7).

Find the vector overrightarrowBA.

  1. 1

    overrightarrowBA goes from B to A, so take B away from A.

  2. 2

    overrightarrowBA=beginpmatrix-4-4 3-7endpmatrix [1]

  3. 3

    =beginpmatrix-8 -4endpmatrix [1]

2. Translation by a vector

A translation is described by a column vector. The top number is the move in the x-direction. The lower number is the move in the y-direction.

To translate a point, add the vector to its coordinates: top number to x, lower number to y. To go back from the image to the starting point, subtract the vector.

To find the vector for a translation, subtract the object's coordinates from the image's.

beginpmatrixx yendpmatrix→beginpmatrixx+a y+bendpmatrix

Examiner insight

“A strategy which would have helped in all parts of this question would have been to draw a sketch of the points and many of these errors may have been avoided.”

Worked example 12 marks

P is the point (–3, 5) and overrightarrowPQ=beginpmatrix8 -3endpmatrix.

Find the coordinates of the point Q.

  1. 1

    Add the vector to the coordinates of P.

  2. 2

    Q=(-3+8, 5+(-3)) [1]

  3. 3

    Q=(5, 2) [1]

3. Adding and subtracting vectors

To add or subtract column vectors, work on the top numbers, then on the lower numbers.

Travelling from A to B and then from B to C is the same as travelling from A to C. The two vectors add to give overrightarrowAC.

Subtracting a negative number adds. Set the working out line by line.

beginpmatrixa bendpmatrix+beginpmatrixc dendpmatrix=beginpmatrixa+c b+dendpmatrix

overrightarrowAB+overrightarrowBC=overrightarrowAC

Examiner insight

“Directed number rules caused quite a few to not score this mark by subtracting a negative number incorrectly.” “However errors were less common when the subtraction of the vectors was set out in the working space.”

Worked example 11 mark

The position vector of A is beginpmatrix4 -1endpmatrix and overrightarrowAB=beginpmatrix-7 5endpmatrix.

Find the position vector of B.

  1. 1

    overrightarrowOB=overrightarrowOA+overrightarrowAB=beginpmatrix4 -1endpmatrix+beginpmatrix-7 5endpmatrix=beginpmatrix-3 4endpmatrix [1]

4. Multiplying a vector by a scalar

A scalar is an ordinary number. To multiply a vector by a scalar, multiply both components by it.

The new vector is parallel to the old one. A positive scalar keeps the direction. A negative scalar reverses the direction.

A fraction is a scalar too. If BQ=2/5BC, then overrightarrowBQ=2/5overrightarrowBC.

kbeginpmatrixa bendpmatrix=beginpmatrixka kbendpmatrix

Examiner insight

“There were more correct responses seen in this part but a noticeable error was not multiplying the second component by 2.” “However, a small number of candidates omitted the – sign or did not multiply the second component by 3.”

Worked example 13 marks

Do not use a calculator. p=beginpmatrix2 -1endpmatrix and q=beginpmatrix-6 8endpmatrix.

Work out 3p+1/2q.

  1. 1

    3p=beginpmatrix6 -3endpmatrix [1]

  2. 2

    1/2q=beginpmatrix-3 4endpmatrix [1]

  3. 3

    3p+1/2q=beginpmatrix6-3 -3+4endpmatrix=beginpmatrix3 1endpmatrix [1]

5. Vectors in terms of a and b

Write the vector you want as a route along vectors you know. Each step along a known vector is added. A step against a known vector is subtracted.

Reversing a vector changes its sign, so overrightarrowBA=-overrightarrowAB. A point part of the way along a line gives a fraction of that vector. If AP : PB = 4 : 1, then overrightarrowAP=4/5overrightarrowAB.

Collect the a terms and the b terms to give the simplest form.

overrightarrowAB=-a+b=b-a when overrightarrowOA=a, overrightarrowOB=b

Examiner insight

“Candidates should be encouraged to show clearer working in vectors questions, particularly showing a route, which would gain credit and focus the candidate on the direction of the vector.” “Clear labelling of intermediate vectors they are trying to find would most likely lead to more marks being earned in many cases.”

Worked example 13 marks

In triangle OAB, overrightarrowOA=a and overrightarrowOB=b. M is the midpoint of AB.

Find overrightarrowOM in terms of a and b. Give your answer in its simplest form.

  1. 1

    route: overrightarrowOM=overrightarrowOA+overrightarrowAM [1]

  2. 2

    overrightarrowAB=b-a, so overrightarrowAM=1/2(b-a) [1]

  3. 3

    overrightarrowOM=a+1/2b-1/2a=1/2(a+b) [1]

Recap

  • Extended: overrightarrowAB is the vector from A to B. Find it as end minus start.
  • A column vector has the x move on top and the y move below.
  • A translation adds its column vector to every point.
  • Add and subtract column vectors component by component.
  • Multiply both components by a scalar. A negative scalar reverses the direction.
  • Find vectors in terms of a and b by writing a route, then simplify.

End-of-chapter exercise

Test yourself on the whole chapter. Work through these before moving on.

  1. Do not use a calculator. Write down the column vector for a move of 3 units left and 5 units up.1 mark
  2. A is the point (2, 5) and B is the point (7, 1). Find overrightarrowAB.2 marks
  3. The point P(–2, 4) is translated by the vector beginpmatrix3 -6endpmatrix. Find the coordinates of its image.2 marks
  4. Do not use a calculator. p=beginpmatrix2 -7endpmatrix and q=beginpmatrix-3 -4endpmatrix. Work out p-q.2 marks
  5. Do not use a calculator. Work out -2beginpmatrix3 -4endpmatrix.2 marks
  6. Do not use a calculator. a=beginpmatrix3 1endpmatrix and b=beginpmatrix-2 4endpmatrix. Work out 2a-3b.3 marks
  7. Do not use a calculator. A is the point (1, –3). overrightarrowAB=beginpmatrix4 5endpmatrix and overrightarrowBC=beginpmatrix-6 2endpmatrix. Find the coordinates of C.3 marks
  8. OABC is a parallelogram. overrightarrowOA=a and overrightarrowOC=c. N is the midpoint of AB. Find overrightarrowON in terms of a and c.3 marks
  9. In triangle OAB, overrightarrowOA=4a and overrightarrowOB=6b. P lies on OA and OP : PA = 3 : 1. Q lies on AB and AQ=2/3AB. Find overrightarrowPQ in terms of a and b. Give your answer in its simplest form.5 marks
  10. Do not use a calculator. The flight of a drone is recorded on a coordinate grid. The drone leaves its base B and flies along 2p-q to a point C. It then flies along -1/2p and lands at the point L (6, -5). p=beginpmatrix4 -6endpmatrix and q=beginpmatrix-3 2endpmatrix. (a) Find overrightarrowBL. (b) Find the coordinates of B. (c) From L the drone flies along kq to a point M on the x-axis. Find the value of k and the coordinates of M.7 marks

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