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Vectors 2

Maths: Edexcel IGCSE Higher, topic 5.1.2

Scalars and Straight Lines

A scalar is any number such as 10, π, -4, 1⁄2, 4.52, √6.

Scalars and vectors

Scalars and vectors

  • When a vector is multiplied by a scalar its direction does not change but its size does.
  • A negative scalar switches the direction of the vector.
  • To multiply a column vector by a scalar multiply both top and bottom numbers by the scalar.
Straight lines

Straight lines

  • To show 3 points A, B and C lie in a straight line show that AB is a scalar multiple of BC.
Ratios and lengths

Ratios and lengths

  • If given side lengths in ratios you can translate this to vector form.
  • If AB : BC = 2 : 3 and A, B, C are in a straight line then 2 AB = 3 BC.

Magnitude of Vectors

Vectors have a magnitude (size) and direction. The magnitude of a vector can be calculated from the column vector.

Calculating the magnitude

Calculating the magnitude

  • You can use Pythagoras' theorem to calculate the magnitude of a vector (xy\large{x}\atop{y}).
  • The magnitude of a vector (xy)({\large{x}\atop{y}}) is x2+y2\sqrt{x^2+y^2}.
Vector notation

Vector notation

  • You will need to be careful when writing vectors in your exam.
  • Vectors will be printed on the exam as AB⃗\vec{AB} or a.
  • The magnitude of a vector will be shown using modulus signs.
  • I.e. magnitude of AB⃗\vec{AB} = |AB⃗\vec{AB}|
  • I.e. magnitude of a = |a|
Using vector notation

Using vector notation

  • In your exam, you will need to show the examiner that you know the difference between the vector a and the letter a.
  • There are two ways to do this:
  • Using an arrow above the vector, e.g. AB⃗\vec{AB}
  • Underlining the letter, e.g. a‾\underline{a}

Proof by Vector Method

Vectors can be used to prove geometric properties of shapes.

Example

Example

  • By method of vectors, prove that STU is a straight line.
Draw the diagram

Draw the diagram

  • Remember to label each points carefully.
Work out the vectors

Work out the vectors

  • Work out the vectors for the points connecting the path from S to T to U, in terms of the vectors p and q.
Work out the vectors

Work out the vectors

Compare

Compare

  • Compare the vector ST with the vector TU.
  • If STU is a straight line, then ST and TU will be multiples of one another with T as a common point.
Compare

Compare

Test yourself on Vectors 2

  • What happens to a vector when it is multiplied by a scalar?

  • What happens to a vector when it is multiplied by a negative scalar?

  • What is the condition for A, B and C lying on a straight line?

  • What is the condition for the points X, Y and Z lying on a straight line?

  • How can you calculate the magnitude of a vector?

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