Year 9 · Chapter 10

Vectors and compound measures

By the end of this chapter you will be able to

Add, subtract and multiply column vectors, and interpret a vector as an arrow
Write the position vector of a point, and show that two vectors are parallel
Use vectors to find a missing vector in a parallelogram or a triangle
Use speed, density and pressure formulae, and rearrange them to find the unknown quantity
Convert speed and density to other units by multiplying by a conversion factor

You should already know

Translations described with a vector (Year 8, Chapter 11)
Rates such as speed, density and price per unit (Year 8, Chapter 4)
Rearranging a simple formula and substituting values (Year 8, Chapter 8, and Year 9, Chapter 4)

Where this leads

Pythagoras and trigonometry with vector magnitudes (Year 9, Chapter 8)
Similar triangles and parallel lines (Year 9, Chapter 9)
Vectors in 2-D geometry and proof in GCSE

National Curriculum for England, Key Stage 3 mathematics, Year 9. Content: Write vectors as column vectors and as arrows, and add, subtract and multiply vectors by a scalar, Find the position vector of a point, and use vectors to show that lines are parallel, Use vectors to solve simple geometric problems in a diagram, Use compound measures such as speed, density and pressure, and rearrange the formula to find any one quantity, Convert between compound units, such as m/s to km/h and g/cm³ to kg/m³.

Exercise P Do you remember?

Non-calculator

FWork out 3 × 4 − 5.

Done

FA car travels 120 km in 2 hours. What is its average speed in km/h?

Done

FWrite 2.5 kg as grams.

Done

FSolve 5x = 35.

Done

FWhich way does an arrow point for a vector (3, 0)?

Done

10.1

Column vectors: adding, subtracting and multiplying

Year 9
Write vectors as column vectors, and add, subtract and multiply vectors by a scalar

Video lesson: Column vectors: adding, subtracting and multiplyingComing soon

A drone flies 3 m right and 2 m up, and then 1 m right and 4 m down. The total movement is found by adding the two column vectors component by component.

A vector has both a size and a direction. A column vector (a / b) moves a point a units across (right if positive) and b units up (down if negative). You can add and subtract vectors, and multiply them by a number, a scalar, by working on each component.

Key ideas

Add column vectors: add the top numbers, then the bottom numbers.
Subtract in the same way: subtract the top, then the bottom.
Multiply by a scalar k: multiply each number in the vector by k.
Negative numbers in a vector mean movement left or down.

Worked example 10.1.1

Work out (3 / 2) + (1 / −4).

Method. Add the column vectors component by component.

Write the vectors. Add the top numbers, then the bottom numbers.
3
2 + 1
−4
Add the components. Top with top and bottom with bottom.
(3 + 1, 2 + −4)
Write the answer as a column vector.
4
−2

Check. Check by counting the arrow steps on a grid, one component at a time.

Watch out. Keep the top and bottom numbers in the right places. Subtracting a negative adds.

Worked example 10.1.2

Work out (5 / −1) − (2 / 3).

Method. Subtract the column vectors component by component.

Write the vectors. Subtract the top numbers, then the bottom numbers.
5
−1 − 2
3
Subtract the components. Top with top and bottom with bottom.
(5 − 2, −1 − 3)
Write the answer as a column vector.
3
−4

Check. Check by counting the arrow steps on a grid, one component at a time.

Watch out. Keep the top and bottom numbers in the right places. Subtracting a negative adds.

Worked example 10.1.3

Work out 3 × (2 / −1).

Method. Multiply the column vectors component by component.

Write the vector. Multiply each part of the vector by 3.
3 × 2
−1
Multiply each part.
(3 × 2, 3 × −1)
Write the answer as a column vector.
6
−3

Check. Check by counting the arrow steps on a grid, one component at a time.

Watch out. Keep the top and bottom numbers in the right places. Subtracting a negative adds.

Watch out

Watch out: subtracting a negative number adds. (5 / −1) − (2 / 3) = (3 / −4), not (3 / 2).

Exam tip

Exam tip. Write each column vector on its own line with a bracket, then work component by component.

10.2

Position vectors and parallel vectors

Year 9
Find the position vector of a point, and show that vectors are parallel

Video lesson: Position vectors and parallel vectorsComing soon

A map has an origin at the school gate. The position vector of a bus stop tells you how far it is across and up from the gate.

The position vector of a point P is the vector from the origin O to P, written as a column vector. Two vectors are parallel if one is a scalar multiple of the other: v = k u.

Key ideas

The position vector of (x, y) is the column vector (x / y).
Vectors are parallel if one is a scalar multiple of the other.
A scalar multiple keeps the same direction, but may change the length.
The midpoint of A and B is the average of their position vectors.

Worked example 10.2.1

A has position vector (2 / 3) and B has position vector (8 / 7). Find the midpoint of AB.

Method. Add the position vectors and divide by 2.

Add the vectors. The midpoint position vector is the average of the two.
(2 + 8, 3 + 7) = (10, 10)
Divide by 2. Halve each component.
(10 / 2, 10 / 2) = (5, 5)
Write the answer.
(5 / 5)

Check. The midpoint is halfway between (2, 3) and (8, 7).

Worked example 10.2.2

Show that the vectors (2 / −4) and (−1 / 2) are parallel.

Method. Find a scalar k with v = k u.

Look for a multiple. Try to write (−1 / 2) as k times (2 / −4).
(−1 / 2) = k (2 / −4)
Find k. From the top: −1 = 2k, so k = −1/2. Check the bottom: −4 × (−1/2) = 2.
k = −1/2
Write the answer. The second vector is −1/2 times the first, so they are parallel.
Parallel: v = −1/2 u

Check. −1/2 × (2 / −4) = (−1 / 2).

Worked example 10.2.3

Point P has position vector (1 / 4) and Q has position vector (5 / 10). Find the vector PQ.

Method. The vector from P to Q is the position of Q minus the position of P.

Write the rule. PQ = OQ − OP.
PQ = (5 / 10) − (1 / 4)
Subtract the components. Subtract the top numbers, then the bottom numbers.
(5 − 1, 10 − 4) = (4, 6)
Write the answer.
(4 / 6)

Check. P plus (4 / 6) gives (5 / 10), which is Q.

Watch out

Watch out: check both components. A vector that is only a multiple of one component is not parallel.

Exam tip

Exam tip. To show two vectors are parallel, write one as a scalar times the other and state the scalar.

Exercise 10B Position vectors and parallel vectors

Non-calculator

FPoint A has position vector 2
3 and point B has position vector 8
7. Find the midpoint of AB.

Done

FPoint A has position vector 0
4 and point B has position vector 6
2. Find the midpoint of AB.

Done

FPoint A has position vector −2
1 and point B has position vector 4
5. Find the midpoint of AB.

Done

FPoint A has position vector 5
−3 and point B has position vector 1
1. Find the midpoint of AB.

Done

FShow that 2
−4 and −1
2 are parallel.

Done

FShow that 3
6 and −1
−2 are parallel.

Done

RAre 2
3 and 4
5 parallel? Explain your answer.

Done

FPoint P has position vector (3 / 1) and point Q has position vector (7 / 5). Find the midpoint of PQ.

Done

10.3

Using vectors in geometry

Year 9
Use vectors to solve simple geometric problems in a diagram

Video lesson: Using vectors in geometryComing soon

A rectangular park ABCD has path AB as a vector a and AD as a vector b. A walker needs the route from A to C, which is a diagonal. The route can be written in terms of a and b by adding the vectors along the sides.

In a diagram, vectors can be written in terms of two basic vectors. Go along the sides and add the vectors: AC = AB + BC = a + b, if BC = b. Reverse a vector to go the other way: BA = −AB.

Key ideas

A vector written in the opposite direction has the opposite sign: BA = −AB.
To go from one point to another, add the vectors along a route.
The diagonal of a parallelogram is the sum of two adjacent sides.
Vectors that are equal have the same length and direction.

Worked example 10.3.1

ABCD is a parallelogram with AB = a and AD = b. Find AC in terms of a and b.

Method. Go from A to C along the sides.

Choose a route. Go from A to B, then from B to C.
AC = AB + BC
Use the parallelogram. BC is equal to AD, so BC = b.
BC = b
Write the answer.
AC = a + b

Check. The other route A to D to C gives AD + DC = b + a, the same answer.

Worked example 10.3.2

In the same parallelogram, find BD in terms of a and b.

Method. Go from B to D through A.

Choose a route. BD = BA + AD.
BD = BA + AD
Use the reverse vector. BA is the opposite of AB, so BA = −a.
BA = −a
Write the answer. AD = b.
BD = b − a

Check. This is the other diagonal, from B to D.

Worked example 10.3.3

In triangle OAB, OA = p and OB = q. M is the midpoint of AB. Find OM in terms of p and q.

Method. Go from O to M through A.

Route. OM = OA + AM, and AM is half of AB.
OM = p + AM
Find AB. AB = AO + OB = −p + q, so AM = (q − p) / 2.
AM = (q − p)/2
Write the answer. OM = p + (q − p)/2 = (p + q)/2.
OM = (p + q)/2

Check. OM is halfway between p and q, as the midpoint should be.

Watch out

Watch out: the reverse of a vector changes its sign. BA is not the same as AB.

Exam tip

Exam tip. Write the route with plus signs, one step at a time, before you substitute the letters.

Exercise 10C Vectors in simple geometric problems

Non-calculator

FABCD is a parallelogram with AB = a and AD = b. Find AC in terms of a and b.

Done

FABCD is a parallelogram with AB = a and AD = b. Find BD in terms of a and b.

Done

FOABC is a parallelogram with OA = p and OC = q. Find AB in terms of p and q.

Done

RThe vector AB is equal to the vector DC in a parallelogram ABCD. Explain why this is true.

Done

FM is the midpoint of AB, and AB = 2a. Find the vector AM in terms of a.

Done

FABC is a triangle with AB = a and AC = b. Find BC in terms of a and b.

Done

FIn a parallelogram ABCD, AB = a and BC = b. Find the vector CD in terms of a.

Done

FP is the point with AP = 3a. Find the vector PA in terms of a.

Done

PSABCD is a parallelogram with AB = a and AD = b. Find the vector from C to A in terms of a and b.

Done

10.4

Speed, density and pressure

Year 9
Use compound measures and rearrange the formula to find any one quantity

Video lesson: Speed, density and pressureComing soon

A runner covers 400 m in 50 s. Her speed is found by dividing the distance by the time. Compound measures such as speed, density and pressure combine two quantities in one formula.

Speed = distance ÷ time. Density = mass ÷ volume. Pressure = force ÷ area. The formula can be rearranged to find any one of the three quantities: for example, distance = speed × time, and time = distance ÷ speed.

Key ideas

Speed = distance / time. Distance = speed × time. Time = distance / speed.
Density = mass / volume. Mass = density × volume. Volume = mass / density.
Pressure = force / area. Force = pressure × area. Area = force / pressure.
Check the units match before you calculate.

Worked example 10.4.1

A runner covers 400 m in 50 s. Find her speed in m/s.

Method. Speed is distance divided by time, with units matching the question.

Write the formula. Speed = distance ÷ time.
speed = distance / time
Substitute. The distance is 400 m and the time is 50 s.
400 ÷ 50
Work out the speed.
= 8 m/s

Check. Check: 8 × 50 = 400.

Watch out. Make sure the units match: a time in minutes must be changed to hours before you use km/h.

Worked example 10.4.2

A block has a mass of 240 g and a volume of 80 cm³. Find its density in g/cm³.

Method. Density is mass divided by volume.

Write the formula. Density = mass ÷ volume.
density = mass / volume
Substitute. The mass is 240 g and the volume is 80 cm³.
240 ÷ 80
Work out the density.
= 3 g/cm³

Check. Check: 3 × 80 = 240.

Watch out. The units of density are mass units per volume unit, for example g/cm³.

Worked example 10.4.3

A car travels at 20 m/s for 15 s. Find the distance travelled.

Method. Rearrange the formula to make distance the subject, then substitute.

Write the formula. The formula is speed = distance / time, so distance = speed × time.
speed = distance / time, so distance = speed × time
Rearrange to make the unknown the subject. Find distance by undoing the formula.
distance = ...
Substitute the values. Put the known values in.
Work out the answer.
300 m

Check. Substitute the answer back into the original formula to check.

Watch out. Rearrange the formula before you substitute. Substituting first is harder to check.

Watch out

Watch out: rearrange the formula before you substitute. Put the numbers in only once you have the unknown on its own.

Exam tip

Exam tip. Write the formula, then the rearranged form, then the substitution. Each line can earn a mark.

10.5

Converting compound units

Year 9
Convert between compound units, such as m/s to km/h and g/cm³ to kg/m³

Video lesson: Converting compound unitsComing soon

A cheetah runs at 25 m/s. To compare it with the speed of a car in km/h, the metres per second must be converted to kilometres per hour.

To change a compound unit, use a conversion factor for each part of the unit. For example, 1 km = 1000 m and 1 hour = 3600 seconds, so 1 m/s = 3.6 km/h. For density, 1 g/cm³ = 1000 kg/m³.

Key ideas

1 m/s = 3.6 km/h, so multiply by 3.6 to go from m/s to km/h.
To go from km/h to m/s, divide by 3.6.
1 g/cm³ = 1000 kg/m³, because 1 cm³ = 0.000001 m³ and 1 g = 0.001 kg.
Check that the size of the answer is sensible.

Worked example 10.5.1

Convert 25 m/s to km/h.

Method. Multiply by the conversion factor to change the unit.

Write the conversion factor. Speed: 1 m/s = 3.6 km/h.
factor = 3.6
Multiply. Multiply the value by the factor, because the new unit is smaller or bigger.
25 × 3.6 = 90
Write the answer with the unit.
90 km/h

Check. Divide 90 by 3.6 to get back to 25 m/s.

Watch out. A bigger unit gives a smaller number, so check that the answer size makes sense.

Worked example 10.5.2

Convert 2.5 g/cm³ to kg/m³.

Method. Multiply by the conversion factor to change the unit.

Write the conversion factor. Density: 1 g/cm³ = 1,000 kg/m³.
factor = 1,000
Multiply. Multiply the value by the factor, because the new unit is smaller or bigger.
2.5 × 1,000 = 2,500
Write the answer with the unit.
2,500 kg/m³

Check. Divide 2,500 by 1,000 to get back to 2.5 g/cm³.

Watch out. A bigger unit gives a smaller number, so check that the answer size makes sense.

Worked example 10.5.3

Convert 72 km/h to m/s.

Method. Multiply by the conversion factor to change the unit.

Write the conversion factor. Speed: 1 km/h = 5/18 m/s.
factor = 5/18
Multiply. Multiply the value by the factor, because the new unit is smaller or bigger.
72 × 5/18 = 20
Write the answer with the unit.
20 m/s

Check. Divide 20 by 5/18 to get back to 72 km/h.

Watch out. A bigger unit gives a smaller number, so check that the answer size makes sense.

Watch out

Watch out: km/h to m/s is divide by 3.6, not multiply. The answer in m/s is smaller than the number of km/h.

Exam tip

Exam tip. Write the conversion factor clearly, then check the answer is about the right size.

Chapter summary

Skill	Method
Column vectors	Add, subtract or scale each component separately.
Position vector	The column vector from the origin to the point.
Parallel vectors	One is a scalar multiple of the other: v = k u.
Vectors in diagrams	Add along a route. Reverse a vector to change its sign.
Speed, density, pressure	Divide the quantity by the other: speed = d / t, density = m / V, pressure = F / A.
Unit conversion	1 m/s = 3.6 km/h. 1 g/cm³ = 1000 kg/m³.

Can you…?

I can add, subtract and multiply column vectors
I can write a position vector and find a midpoint
I can show that two vectors are parallel
I can use vectors to find routes in a parallelogram or triangle
I can use speed, density and pressure, and rearrange the formula
I can convert speed and density to other units

Key terms

vector: a quantity with both size and direction, written as a column vector or an arrow
scalar: a number that multiplies a vector and changes its size, but not its direction
position vector: the vector from the origin to a point
compound measure: a measure made from two quantities, such as speed or density
conversion factor: a number that changes a value from one unit to another

Total 23 marks. Allow about 23 minutes.

Vectors and compound measures

By the end of this chapter you will be able to

You should already know

Where this leads

Exercise P Do you remember?

Column vectors: adding, subtracting and multiplying

Key ideas

Watch out

Exam tip

Exercise 10A Adding, subtracting and multiplying vectors

Position vectors and parallel vectors

Key ideas

Watch out

Exam tip

Exercise 10B Position vectors and parallel vectors

Using vectors in geometry

Key ideas

Watch out

Exam tip

Exercise 10C Vectors in simple geometric problems

Speed, density and pressure

Key ideas

Watch out

Exam tip

Exercise 10D Speed, density and pressure

Converting compound units

Key ideas

Watch out

Exam tip

Exercise 10E Converting compound units

Chapter summary

Can you…?

Key terms

Exercise 10Z Check-up

Exercise 10X Practice test questions