19.6 Vectors 1

A free GCSE Maths Foundation lesson for Edexcel (9–1) from OBA School: column vectors, vector notation, adding and subtracting vectors, and vector journeys. Watch the video, read the explanation and worked examples, then try the revision questions. You can work through the full lesson, with interactive activities, practice quizzes and printable worksheets, in our free online course.

What you will learn

• Write a movement on a grid as a column vector and use it to translate points
• Use the notations and a, and recognise equal and negative vectors
• Add and subtract column vectors, and show the result on a diagram
• Describe a journey between two points as a sum of vectors

Key words

• Vector: A quantity with a size (length) and a direction, such as a movement or a force.
• Column vector: Two numbers written in a column, such as . Top: across (+ right, − left). Bottom: up (+) or down (−).
• Resultant: The single vector that has the same effect as two or more vectors one after another.
• Equal vectors: Vectors with the same length and the same direction. They can start in different places.

Column vectors

A column vector tells you how far to move across and then up or down. The vector from P to Q in the diagram is = , and the vector from R to S is = .

Worked example: translating a point

Translate the point (2, 5) by the vector .
Add the top number to x and the bottom number to y: (2 + 3, 5 − 4)

Answer: (5, 1)

Worked example: vector between two points

A is (−1, 3) and B is (4, −2). Find .
Across: 4 − (−1) = 5. Up: −2 − 3 = −5

Answer:

Naming vectors, equal and negative vectors

A vector can be named by its start and end points, , or by a bold letter, a. When you write by hand, underline it: a.

• Equal vectors have the same length and direction, wherever they are drawn. • −a has the same length as a but the opposite direction. Both numbers change sign: if a = then −a = . • = −: going backwards along a vector changes its sign.

Worked example: negative vector

= . Write as a column vector.
Change the sign of both numbers.

Answer:

Adding and subtracting vectors

To add column vectors, add the top numbers and add the bottom numbers. Subtracting works the same way. On a diagram, a + b means "go along a, then along b" (nose to tail).

Worked example: adding and subtracting

a = and b = .
a + b = =
a − b = =

Answer: and

Key fact: Take care with negatives: subtracting a negative number is the same as adding.

Vector journeys

To get from one point to another, follow any route along known vectors. + = . If you travel against an arrow, use a minus sign.

Worked example: a journey through O

There is no direct vector, so go from A back to O, then out to B.
= + = −a + b

Answer: = b − a

Common mistakes

• Writing the column vector upside down. The top number is always the move across.
• Getting the signs wrong: left and down are negative.
• Forgetting that = − when you travel the opposite way along a vector.
• Adding the top number to the bottom number. Add top to top and bottom to bottom.

Related lessons

• Previous lesson: 19.5 Congruence 2
• Next lesson: 19.7 Vectors 2
• In this chapter: 19.1 Similarity and Enlargement
• In this chapter: 19.2 More Similarity
• In this chapter: 19.3 Using Similarity

Continue in the full course

This lesson is section 19.6 (EGCMF1906) of our free GCSE Maths Foundation course. In the full course you get every video for this section, an animated interactive lesson, a self-marking practice quiz, a printable worksheet with a mark scheme, and your progress saved. Go to the full GCSE Maths Foundation course.