19.7 Vectors 2

A free GCSE Maths Foundation lesson for Edexcel (9–1) from OBA School: multiplying a vector by a scalar, combining vectors and parallel vectors. 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

• Multiply a vector by a number (a scalar), including negative numbers and fractions
• Work out combinations such as 2a + 3b and 3a − 2b
• Decide whether two vectors are parallel
• Write vectors on a diagram in terms of a and b, and find unknowns in vector equations

Key words

• Scalar: An ordinary number, with size but no direction. Multiplying a vector by a scalar changes its length.
• Multiple: ka is a multiple of a. If a = then 3a = .
• Parallel: Vectors pointing in the same (or exactly opposite) direction. One is a multiple of the other.

Multiplying by a scalar

Multiply both numbers of the column vector by the scalar. 2a is twice as long as a in the same direction. −2a is twice as long in the opposite direction. (1/2)a is half as long.

Worked example: scalar multiples

a = .
3a = = −2a = (1/2)a =

Answer: , and

Combining vectors

Work out each multiple first, then add or subtract the top numbers and the bottom numbers.

Worked example: 3a − 2b

a = and b = .
3a = and 2b =
3a − 2b = =

Answer:

Key fact: Brackets work as usual: 2(a + b) = 2a + 2b. You can simplify expressions such as 3a + b − a = 2a + b.

Parallel vectors

Two vectors are parallel if one is a number times the other. Divide matching numbers: if you get the same answer for the top and the bottom, they are parallel.

Worked example: parallel or not?

and : 6 ÷ 2 = 3 and −9 ÷ −3 = 3. Same multiplier, so parallel.
and : 6 ÷ 4 = 1.5 but 8 ÷ 6 = 1.33… Different, so not parallel.

Answer: The first pair is parallel; the second is not

Vectors on a diagram and unknowns

On a grid of parallelograms, every move along a side is a or b (or their negatives). Count the steps to write any vector in terms of a and b.

Worked example: reading the diagram

= 3a + 2b (the order of the steps does not matter)

Answer: 3a + 2b

Worked example: finding unknowns

Find k and m when 2 + = .
Top: 6 + m = 4, so m = −2. Bottom: 2k + 1 = 9, so k = 4.

Answer: k = 4, m = −2

Common mistakes

• Multiplying only the top number by the scalar. Both numbers must be multiplied.
• Losing a minus sign: −3 × −2 = +6.
• Saying vectors are parallel because they "look similar". Check that the same multiplier works for both numbers.
• Thinking −2a is not parallel to a. It is parallel, just pointing the opposite way.

Related lessons

• Previous lesson: 19.6 Vectors 1
• Next lesson: 20.1 Graphs of Cubic and Reciprocal Functions
• 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 19.7 (EGCMF1907) 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.