Chapter 1 of 8

All 8 chapters
  1. 1Scalars and vectors, readunderstood
  2. 2Velocity and acceleration, readunderstood
  3. 3Newton's first law, readunderstood
  4. 4Newton's second law, readunderstood
  5. 5Weight and falling, readunderstood
  6. 6Newton's third law, readunderstood
  7. 7Momentum, readunderstood
  8. 8Stopping distances, readunderstood

S4.8.1 Chapter 1

Scalars and vectors

Mei walked 3 km east and then 3 km west. Did she go 6 km, or did she go nowhere?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

  • You picked A. Read on. At the end, the chapter comes back to your guess.

  • You picked B. Read on. At the end, the chapter comes back to your guess.

  • You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

  • Sort quantities into scalars and vectors
  • Find the resultant of two forces along a line
  • Add two forces at right angles with a scale drawing

Some quantities need a number and a unit. Others need a direction too. This difference matters for every law in this book.

Mei walks 3 km east to the market. Then she walks 3 km west and stands at her own door. She walked a long way. Yet she is not away from home at all.

A number line for distance east of her door, in kilometres, with the numbers 0, 1, 2, 3 and 4. A top arrow runs from 0 to 3 and is labelled 3 km east. A lower arrow runs from 3 back to 0 and is labelled 3 km west. The axis is named km east of her door.01234km east of her door3 km east3 km west
The two arrows are the two parts of the walk. Each has a size and a direction.

The distance she walked is 3 km + 3 km = 6 km. Distance counts every step, in any direction. It has a size and no direction.

The displacement is how far she is from where she began, in a straight line, and in which direction. Here it is 0 km. Her two walks cancel.

A quantity with a size and no direction is a scalar. A quantity with a size and a direction is a vector. Distance is a scalar. Displacement is a vector.

A table with two columns, Scalar and Vector. The rows pair distance with displacement, speed with velocity, mass with weight, time with acceleration, and temperature with force.ScalarVectordistancedisplacementspeedvelocitymassweighttimeaccelerationtemperatureforce
The first three rows pair a scalar with the vector that goes with it. The last two rows are one more scalar and one more vector. Weight is a force, and a force has a direction.

To give a vector, say its size, its unit and its direction: "20 N to the left". We draw it as an arrow. The length of the arrow shows the size. The head shows the direction.

We often pick one direction as positive, say right. A vector to the left then has a minus sign. The minus sign shows only the direction. A force of −15 N has the same size as +15 N and acts the other way.

Forces are vectors, so we add them with their directions. Start with forces on one line. Ravi pulls a box to the right with 300 N. Amina pulls it to the left with 200 N.

Draw it first

Two forces on one line

  1. A box with two pulls on it, drawn to the same scale. A bar marked bar = 100 N shows the scale. First an arrow to the right labelled Ravi, 300 N. Then an arrow to the left labelled Amina, 200 N. Last both arrows are replaced by one arrow to the right labelled resultant, 100 N.Ravi300 Nbar = 100 N

    1 Ravi pulls to the right with 300 N. The arrow is 300 N long on the scale bar.

  2. A box with two pulls on it, drawn to the same scale. A bar marked bar = 100 N shows the scale. First an arrow to the right labelled Ravi, 300 N. Then an arrow to the left labelled Amina, 200 N. Last both arrows are replaced by one arrow to the right labelled resultant, 100 N.Ravi300 NAmina200 Nbar = 100 N

    2 Amina pulls to the left with 200 N. Her arrow points the other way and is shorter.

  3. A box with two pulls on it, drawn to the same scale. A bar marked bar = 100 N shows the scale. First an arrow to the right labelled Ravi, 300 N. Then an arrow to the left labelled Amina, 200 N. Last both arrows are replaced by one arrow to the right labelled resultant, 100 N.resultant100 Nbar = 100 N

    3 Take 200 N from 300 N. One force of 100 N to the right does the work of both. It is the resultant.

The resultant is the single force that has the same effect as all the forces together. On one line, forces in the same direction add. Forces in opposite directions subtract. The resultant points the way of the bigger force.

If the two forces are equal and opposite, the resultant is 0 N. The forces are balanced.

See it three ways

As a picture

A block with an arrow to the right labelled push, 400 N and a shorter arrow to the left labelled friction, 120 N. A bar marked bar = 200 N shows the scale.push400 Nfriction120 Nbar = 200 N

In words

On one line, forces in the same direction add. Forces in opposite directions subtract. The resultant points the way of the bigger force.

In symbols

400 N − 120 N = 280 N, to the right.

Worked example

Kofi pushes a cart to the right with 250 N. Lucía pushes it to the right with 140 N. Friction pushes it to the left with 120 N. Find the resultant force.

  1. Choose right as the positive direction.
  2. The forces to the right add: 250 N + 140 N = 390 N.
  3. The force to the left is 120 N.
  4. Resultant = 390 N − 120 N = 270 N, to the right.

Answer: 270 N to the right

Now put the forces at right angles. Hana pulls a sledge east with 30 N. Diego pulls it north with 40 N. East and north are at 90° to each other. The sledge goes neither east nor north.

We cannot add 30 N and 40 N as plain numbers. We draw them. Put the second arrow at the head of the first arrow, as two steps in a walk. The resultant runs from the start to the end.

A grid where one square is 10 N. A thin arrow 3 squares long to the right is labelled Hana 30 N. From its head a thin arrow 4 squares long goes up and is labelled Diego 40 N. A thick arrow from the start to the end is labelled resultant 50 N. A small square mark shows the right angle between the two forces. The scale is written 1 square = 10 N.Hana 30 NDiego 40 Nresultant 50 N1 square = 10 N
The thick arrow is the resultant. A ruler on the grid shows it is 5 squares long.

Worked example

Hana pulls a sledge east with 30 N. Diego pulls it north with 40 N. Find the size of the resultant with a scale drawing.

  1. Let 1 square stand for 10 N. The 30 N force is 3 squares. The 40 N force is 4 squares.
  2. Draw the first arrow 3 squares east. From its head draw the second arrow 4 squares north.
  3. Draw the resultant from the start to the end. Measure it: 5 squares.
  4. Resultant = 5 squares × 10 N = 50 N.

Answer: 50 N

Pythagoras gives the same size without a drawing. The resultant is the longest side of a right angled triangle. So 30² + 40² = 900 + 1 600 = 2 500, and the square root of 2 500 is 50. A scale drawing is a check, and it also shows the direction. Here the resultant points about 53° north of east. A protractor on the drawing gives this angle.

At right angles the resultant is always less than the two sizes added. 50 N is less than 30 N + 40 N. Only forces on one line, in the same direction, add straight up.

Try it

Walk a right angle

You need: two small stones, a piece of string, about 8 steps long, flat open ground.

Be safe: Choose flat, open ground away from traffic. Walk slowly and watch where you step.

  1. Put one stone down. This is your start.
  2. Walk 4 steps straight ahead. Try to keep every step the same length.
  3. Turn to your right, a quarter turn. Walk 3 steps.
  4. Put the second stone down. Lay the string straight from the first stone to the second.
  5. Count how many steps long the string is.

What you will see: You walked 4 + 3 = 7 steps. The string is only about 5 steps long. The string shows your displacement. It is the resultant of the two walks, and it is shorter than the sum.

If you cannot try it, this is what happens.
A table with the columns Part and Length. Ahead is 4 steps. Right is 3 steps. Total walked is 7 steps. String is 5 steps.PartLengthahead4 stepsright3 stepstotal walked7 stepsstring5 steps

A wrong idea many people have

Forces of 30 N and 40 N always add up to 70 N.

Why it feels right: We add numbers all the time, and 30 + 40 is 70.

The test that breaks it: Draw the two forces at right angles on a grid, head to tail. The resultant measures 5 squares, which is 50 N. The walk in the Try it gives 7 steps walked and 5 steps from the start.

What is true: The sizes add to 70 N only when both forces act on one line in the same direction. At right angles, the resultant is the third side of a triangle, and it is shorter.

A wrong idea many people have

A displacement of −3 km means a shorter trip than +3 km.

Why it feels right: In everyday sums, −3 is less than +3.

The test that breaks it: Start at the door and walk 3 km east. Another day, start at the same door and walk 3 km west. Each walk ends 3 km from the door, on opposite sides.

What is true: The minus sign tells only the direction. Both walks have a size of 3 km. A displacement of −3 km is 3 km the other way.

Key words

distance
how far something travels in all, along its path. It is a scalar.
displacement
the straight line from the start to the end, with its direction. It is a vector.
scalar
a quantity with a size and no direction.
vector
a quantity with a size and a direction.
resultant
the one force that has the same effect as all the forces on a thing.

Back to your guess

The answer is C: Both are right. They are two different quantities.

Her distance walked is 6 km. Her displacement from the start is 0 km. Distance has a size only. Displacement has a size and a direction. Both are true at once.

Check yourself

Try each one before you look. A wrong answer tells you where to go next.

  1. Which of these is a vector?

    Tap your answer.

    1. Not this one. Mass has a size and no direction. It is a scalar. Go to S3.7.4. Try another.

    2. Not this one. Time has a size and no direction. It is a scalar. Try another.

    3. Yes, that is right. A push is a force. A force has a size and a direction, so it is a vector.

    4. Not this one. A temperature says how hot, not which way. It is a scalar. Try another.

    See the answer

    A push to the left

    A push is a force. A force has a size and a direction, so it is a vector.

    • If you chose A: Mass has a size and no direction. It is a scalar. Go to S3.7.4.
    • If you chose B: Time has a size and no direction. It is a scalar.
    • If you chose D: A temperature says how hot, not which way. It is a scalar.

  2. Tariq walks along a straight road. He goes 6 km east and then 2 km west. What is his displacement from his start?

    Tap your answer.

    1. Not this one. You added the two distances. The second part of the walk goes the other way, so it takes away. Try another.

    2. Yes, that is right. East is positive: 6 km + (−2 km) = 4 km. He ends 4 km east of his start. He walked 8 km in all, which is the distance.

    3. Not this one. The bigger part of the walk was east, so he ends east of his start. Try another.

    4. Not this one. That is the distance walked, with no direction. A displacement must have a direction. Try another.

    See the answer

    4 km east

    East is positive: 6 km + (−2 km) = 4 km. He ends 4 km east of his start. He walked 8 km in all, which is the distance.

    • If you chose A: You added the two distances. The second part of the walk goes the other way, so it takes away.
    • If you chose C: The bigger part of the walk was east, so he ends east of his start.
    • If you chose D: That is the distance walked, with no direction. A displacement must have a direction.

  3. Ravi pulls a box to the right with 70 N. Amina pulls it to the left with 45 N. What is the resultant force?

    A box with an arrow to the right labelled Ravi, 70 N and a shorter arrow to the left labelled Amina, 45 N. A bar marked bar = 20 N shows the scale.Ravi70 NAmina45 Nbar = 20 N
    Tap your answer.

    1. Not this one. You added the two sizes. The forces act in opposite directions, so they take away. Try another.

    2. Not this one. The size is right. The bigger force, 70 N, is to the right, so the resultant points right. Try another.

    3. Not this one. That leaves out Amina. Her pull takes some of Ravi’s pull away. Try another.

    4. Yes, that is right. 70 N − 45 N = 25 N. The bigger force points right, so the resultant points right.

    See the answer

    25 N to the right

    70 N − 45 N = 25 N. The bigger force points right, so the resultant points right.

    • If you chose A: You added the two sizes. The forces act in opposite directions, so they take away.
    • If you chose B: The size is right. The bigger force, 70 N, is to the right, so the resultant points right.
    • If you chose C: That leaves out Amina. Her pull takes some of Ravi’s pull away.

  4. The wind pushes a boat east with 90 N. The current pushes it north with 120 N. What is the size of the resultant force?

    A grid where one square is 30 N. A thin arrow 3 squares long to the right is labelled wind 90 N. From its head a thin arrow 4 squares long goes up and is labelled current 120 N. A small square mark shows the right angle. The scale is written 1 square = 30 N.wind 90 Ncurrent 120 N1 square = 30 N
    Tap your answer.

    1. Yes, that is right. The forces are at right angles, so the resultant is the longest side of a right angled triangle. 90² + 120² = 8 100 + 14 400 = 22 500. The square root of 22 500 is 150 N.

    2. Not this one. You added the sizes. That works only for forces on one line. At right angles the resultant is shorter. Try another.

    3. Not this one. You took one size from the other. Subtracting is for opposite forces, not for forces at right angles. Try another.

    4. Not this one. That is the mean of the two sizes. The resultant is the third side of the triangle, not a mean. Try another.

    See the answer

    150 N

    The forces are at right angles, so the resultant is the longest side of a right angled triangle. 90² + 120² = 8 100 + 14 400 = 22 500. The square root of 22 500 is 150 N.

    • If you chose B: You added the sizes. That works only for forces on one line. At right angles the resultant is shorter.
    • If you chose C: You took one size from the other. Subtracting is for opposite forces, not for forces at right angles.
    • If you chose D: That is the mean of the two sizes. The resultant is the third side of the triangle, not a mean.

  5. Which pair is a scalar first and then a vector?

    Tap your answer.

    1. Not this one. It is the other way round. Velocity has a direction, so it is the vector. Try another.

    2. Not this one. Force is a vector and mass is a scalar, so this pair is back to front. Try another.

    3. Not this one. Both are forces, and a force is a vector. Neither is a scalar. Try another.

    4. Yes, that is right. Distance has a size only, so it is a scalar. Displacement has a size and a direction, so it is a vector.

    See the answer

    distance, displacement

    Distance has a size only, so it is a scalar. Displacement has a size and a direction, so it is a vector.

    • If you chose A: It is the other way round. Velocity has a direction, so it is the vector.
    • If you chose B: Force is a vector and mass is a scalar, so this pair is back to front.
    • If you chose C: Both are forces, and a force is a vector. Neither is a scalar.

  6. Take right as positive. A force of −15 N acts on a box. What does the minus sign tell you?

    Tap your answer.

    1. Not this one. The size is the same. Only the direction differs. Try another.

    2. Yes, that is right. On one line, a minus sign shows the direction opposite to the one chosen as positive. The size is still 15 N.

    3. Not this one. A force is always a vector. The minus sign is how we write its direction on one line. Try another.

    See the answer

    The force acts to the left, the opposite way to positive.

    On one line, a minus sign shows the direction opposite to the one chosen as positive. The size is still 15 N.

    • If you chose A: The size is the same. Only the direction differs.
    • If you chose C: A force is always a vector. The minus sign is how we write its direction on one line.

  7. Two forces of 5 N and 12 N act at right angles. What is the size of the resultant?

    See the answer

    13 N

    5² + 12² = 25 + 144 = 169. The square root of 169 is 13 N. A scale drawing with 1 square for 1 N gives a resultant 13 squares long.

  8. A sledge is pulled to the right with 90 N and to the left with 35 N. What is the resultant force?

    See the answer

    55 N to the right

    The forces are on one line and opposite. 90 N − 35 N = 55 N. The bigger force points right, so the resultant points right.

Keep it

Copy this into your exercise book.

A scalar has a size only. A vector has a size and a direction. On one line, forces add or take away. At right angles they make a triangle: 30 N and 40 N give 50 N.

Draw: A right angled triangle of arrows, 3 squares and 4 squares, with the resultant of 5 squares.

Reading with a child? A short page for the adult: what this chapter teaches and three questions to ask.

No internet tomorrow? Save the whole book as one file (825 KB). It opens on any phone.