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Cambridge IGCSE0607

Vectors in two dimensions

International Mathematics 0607 Chapter Notes

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Vectors in two dimensions
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In this chapter

  1. 1Column vectors and the vector AB
  2. 2Describing a translation
  3. 3Adding and subtracting vectors
  4. 4Multiplying a vector by a scalar
  5. 5Using vectors to find a point

1. Column vectors and the vector AB

A column vector beginpmatrixx yendpmatrix gives a movement: x units right (left if negative) and y units up (down if negative). A column vector has no fraction line between its two numbers. A vector is printed as overrightarrowAB or a.

overrightarrowAB is the vector from A to B. Subtract the coordinates of the start from the coordinates of the end, end minus start.

overrightarrowBA is the opposite direction, so it has the opposite signs.

overrightarrowAB = end - start = beginpmatrixx_B - x_A y_B - y_Aendpmatrix

overrightarrowBA = -overrightarrowAB

Examiner insight

“Some candidates gave the reverse vector. A very small number of candidates reversed the components.” “A few used fraction lines between the vectors entries – this is not correct.”

Worked example 12 marks

A is the point (-3, 2) and B is the point (4, -7).

Find the column vector overrightarrowAB.

  1. 1

    end minus start: x = 4 - (-3) = 7 [1]

  2. 2

    y = -7 - 2 = -9, so overrightarrowAB = beginpmatrix7 -9endpmatrix [1]

2. Describing a translation

A translation moves every point of a shape by the same column vector. The top number is the movement right, the bottom number is the movement up.

To describe it fully, write the word translation and give the vector. Each earns a mark. Coordinates are not accepted in place of the vector.

To find the vector, subtract a point from its image: image minus object.

image = object + beginpmatrixx yendpmatrix

beginpmatrixx yendpmatrix = image - object

Examiner insight

“The word translate, or translation, was required with words such as translocation, shift and move not accepted. A vector was also required, with answers given in coordinate form not accepted.” “The most common mistake was to make a sign error in the vector.”

Worked example 12 marks

Triangle T has vertices (3, 4), (5, 4) and (3, 7). Triangle U has vertices (1, 9), (3, 9) and (1, 12).

Describe fully the single transformation that maps triangle T onto triangle U.

  1. 1

    every vertex moves the same way, so it is a translation [1]

  2. 2

    image minus object for one vertex: beginpmatrix1-3 9-4endpmatrix = beginpmatrix-2 5endpmatrix [1]

3. Adding and subtracting vectors

To add vectors, add the top numbers and add the bottom numbers. To subtract, take the top from the top and the bottom from the bottom.

Write the answer as a column vector. Take care with a negative number being subtracted: taking away -4 is the same as adding 4.

Adding vectors follows the journey: go along a, then along b, and the single vector from the start to the finish is a + b.

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

beginpmatrixa bendpmatrix - beginpmatrixc dendpmatrix = beginpmatrixa - c b - dendpmatrix

Worked example 12 marks

p = beginpmatrix3 -2endpmatrix and q = beginpmatrix-4 5endpmatrix.

Find p - q.

  1. 1

    top: 3 - (-4) = 7 [1]

  2. 2

    bottom: -2 - 5 = -7, so p - q = beginpmatrix7 -7endpmatrix [1]

4. Multiplying a vector by a scalar

A scalar is an ordinary number. To multiply a vector by a scalar, multiply both numbers in the column by it.

A negative scalar changes the sign of both numbers. A fraction such as 1/2 halves both numbers.

In a mixed question, do the scalar multiplications first, then add or subtract.

kbeginpmatrixx yendpmatrix = beginpmatrixkx kyendpmatrix

-beginpmatrixx yendpmatrix = beginpmatrix-x -yendpmatrix

Examiner insight

“The majority of candidates were able to answer this part. For others there seemed to be confusion about how to deal with the 2 outside the second vector.”

Worked example 12 marks

a = beginpmatrix6 -4endpmatrix and b = beginpmatrix1 2endpmatrix.

Find 1/2a + b.

  1. 1

    1/2a = beginpmatrix3 -2endpmatrix [1]

  2. 2

    beginpmatrix3 -2endpmatrix + beginpmatrix1 2endpmatrix = beginpmatrix4 0endpmatrix [1]

5. Using vectors to find a point

A vector locates a point. If overrightarrowPR = koverrightarrowPQ, then R is found by adding the vector to the coordinates of P, the start of the vector.

The answer to a question about a point is a coordinate pair (x, y), not a column vector. Write it in brackets in a row.

A sketch of the points helps you check that the answer is the right size and in the right direction.

point = start + vector

Examiner insight

“Some candidates had difficulty knowing that the top value in the vector went with the x co-ordinate and the lower one with the y co-ordinate.”

Worked example 13 marks

P is the point (3, 7) and Q is the point (7, 4). overrightarrowPQ = beginpmatrix4 -3endpmatrix.

overrightarrowPR = 2overrightarrowPQ

Find the coordinates of R.

  1. 1

    2overrightarrowPQ = beginpmatrix8 -6endpmatrix [1]

  2. 2

    x = 3 + 8 = 11 [1]

  3. 3

    y = 7 + (-6) = 1, so R is (11, 1) [1]

Recap

  • A column vector gives the movement: top number right, bottom number up.
  • overrightarrowAB = end minus start. overrightarrowBA has the opposite signs.
  • A translation is described by the word translation and its vector.
  • Vector = image minus object.
  • Add or subtract vectors top with top, bottom with bottom.
  • Multiply a vector by a scalar by multiplying both numbers.
  • A point is start plus vector, and is written as coordinates.

End-of-chapter exercise

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

  1. Do not use a calculator. A is the point (2, 5) and B is the point (6, 1). Find the column vector overrightarrowAB.2 marks
  2. Do not use a calculator. Triangle T has vertices (4, -3), (6, -3) and (4, 0). Triangle U has vertices (1, 2), (3, 2) and (1, 5). Describe fully the single transformation that maps triangle T onto triangle U.2 marks
  3. A translation has vector beginpmatrix-2 3endpmatrix. Find the image of the point (5, -1).2 marks
  4. Do not use a calculator. a = beginpmatrix3 -1endpmatrix and b = beginpmatrix-2 4endpmatrix. Find a - b.2 marks
  5. c = beginpmatrix-2 5endpmatrix. Find -2c.2 marks
  6. Do not use a calculator. p = beginpmatrix4 -3endpmatrix and q = beginpmatrix-1 2endpmatrix. Find 2p + 3q.3 marks
  7. p = beginpmatrix5 1endpmatrix and q = beginpmatrix2 -3endpmatrix. Find p - 2q.3 marks
  8. Do not use a calculator. A is the point (1, 2) and B is the point (4, 8). overrightarrowAC = 2/3overrightarrowAB. Find the coordinates of C.3 marks
  9. P is the point (5, -2). overrightarrowQP = beginpmatrix-3 4endpmatrix and overrightarrowQR = 2overrightarrowQP - beginpmatrix1 -5endpmatrix. Find the coordinates of R.4 marks
  10. A warehouse robot moves on a floor grid measured in metres. In each column vector the top number is the distance east and the bottom number is the distance north. The robot starts at S (-4, 5). It makes k moves of a = beginpmatrix3 -2endpmatrix, then 2 moves of b = beginpmatrix-1 4endpmatrix, and finishes at F (9, p). Find k, p and the single vector that takes the robot from F straight back to S.5 marks

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