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SQA National 5 Mathematics

Vector notation

Writing and interpreting vectors using components and directed line segments.

Formula referenceMixed revision

Formula reference

Check the National 5 rules and formulae linked to this topic.

Which method?

Match exam clues to a suitable method.

Before you start

  • Know that vectors have size and direction.
  • Use bold or underlined letters in written work.
  • Remember that reversing direction changes the sign.
National 5 Mathematics lesson

Explanation

A vector describes a movement. For example, vector AB describes the movement from A to B.

Column vectors show horizontal and vertical movement. The top number is horizontal, the bottom number is vertical.

Position vectors start from the origin, so if OA = a and OB = b, then AB = b − a

Visual support

Vector as movement from A to B

ABAB = b - a

Key formulae and rules

  • AB = b − a when OA = a and OB = b
  • column vector (x, y) means x across and y up
  • -a is vector a in the opposite direction

Watch out

Writing AB as a − b instead of b − a.

Check

Check direction and length separately. A correct vector needs both components in the right order.

Exam tip

For a vector from one point to another, destination minus start is the safest rule.

Calculator tip

Use brackets for fractions, powers and square roots, then round only at the final line.

Worked examples

Worked example 1

Use position vectors

OA = a and OB = b. Write AB

  1. Move from A back to O: −a.
  2. Move from O to B: b.
  3. Combine the path: −a + b.

Answer: AB = b − a

Worked example 2

Use a column vector

A point moves 4 right and 3 down. Write the column vector.

  1. Horizontal movement is +4.
  2. Vertical movement is −3.
  3. Place horizontal movement above vertical movement.

So: The vector is (4, −3).

Worked example 3

Find a vector between coordinates

A(2, 5) and B(9, 1). Find AB.

  1. Subtract coordinates: B − A.
  2. Horizontal: 9 − 2 = 7
  3. Vertical: 1 − 5 = −4

Answer: AB = (7, −4)

Watch out

  • Writing AB as a − b instead of b − a.
  • Mixing up horizontal and vertical components.
  • Forgetting that a negative component means movement left or down.