Year 11 Physics Module 1 ⏱ ~30 min 5 MC · 3 Short Answer Lesson 1 of 8

Scalars and Vectors, Quantities with Direction

A drone delivering pathology samples in regional NSW flies 6 km from the clinic, then 6 km back. Its odometer logs 12 km, but to the GPS, it has gone nowhere. One number ignored direction; the other did not. That single difference is the foundation of all of kinematics.

Today's hook: The drone's odometer reads 12 km but its displacement is zero. How can two "distance" numbers for the same trip be so different?
0/5TASKS
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Orient and predict

Before you read, predict

A student walks 50 m east to the canteen, then 50 m back west to the classroom. A friend asks two questions: "How far did you walk?" and "How far are you now from where you started?" Why are the two answers different, and which answer needed you to track direction?

Which of the following quantities tells you both how much and in which direction?

Learning Intentions
goals

Know

  • The definition of a scalar and a vector quantity
  • Which motion quantities are scalars (distance, speed) and which are vectors (displacement, velocity, acceleration)
  • How a vector is represented as an arrow, length and direction
  • What equal vectors and negative vectors are
  • The role of a sign convention (+/−) on a line

Understand

  • Why direction makes a vector behave differently from a scalar
  • Why two trips with the same distance can have different displacements
  • Why vectors along a line must be added with regard to sign
  • Why a negative answer is a direction, not "less than nothing"

Can Do

  • Classify any motion quantity as scalar or vector
  • Draw a vector to scale as a labelled arrow
  • Set a sign convention and add/subtract vectors in one dimension
  • Write a vector with its magnitude and direction
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Scan the key terms

Scan these before reading
vocab
Scalara quantity with magnitude (size) only, no direction; e.g. distance, speed, time, mass, temperature
Vectora quantity with both magnitude and direction; e.g. displacement, velocity, acceleration, force
Magnitudethe size of a quantity, always a positive number with a unit (e.g. 12 km)
Resultantthe single vector that has the same effect as two or more vectors added together
Equal vectorstwo vectors with the same magnitude AND the same direction, even if drawn in different places
Negative vectora vector with the same magnitude but the opposite direction to a given vector
Sign conventiona chosen rule (e.g. east = +, west = −) that lets directions along a line be written with + and − signs
Cross-lesson links: L01 is the foundation of all of Kinematics. The scalar/vector distinction drilled here splits every later quantity into two columns: distance vs displacement (L02), speed vs velocity (L03), and acceleration as a vector (L04). The sign convention you learn here is reused in motion graphs (L05), the equations of motion (L06), and full 2D vector resolution (L07).
Misconceptions to fix
A negative vector means a negative amount, like "less than zero metres".A minus sign on a vector is a direction label, not a smaller size. −5 m means 5 m in the negative direction; its magnitude is still 5 m.
A longer arrow always means a "bigger" or "more important" quantity.Arrow length encodes magnitude to a chosen scale, and the arrowhead encodes direction. Both parts carry meaning, a short arrow pointing the right way can matter more than a long one.

True or false: a displacement of −8 m has a smaller magnitude than a displacement of +3 m, because −8 is less than +3.

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Split every quantity in two

Core Content

01
Scalars and Vectors, The Big Split
+5 XP

Some measurements are complete with a single number and a unit. "The water is 21 °C." "The bag weighs 1.2 kg." Nobody asks "21 °C in which direction?", the question makes no sense. But other measurements are only half-finished without a direction. "The wind is 30 km/h" leaves out the most important thing for a sailor: which way? Physics sorts every quantity into these two boxes.

A scalar is a quantity that has magnitude only a size with a unit. A vector is a quantity that has both magnitude and direction. The magnitude of a vector is always a positive number; the direction is extra, essential information.

Sorting the motion quantities

Scalars (magnitude only)

Distance (how far the path is), speed (how fast), time, mass, temperature, energy. A bare number with a unit fully describes them.

Vectors (magnitude + direction)

Displacement (how far and which way from start), velocity (speed in a direction), acceleration, and force. Leave out the direction and the description is incomplete.

Australian context: A weather bureau warning of "winds to 90 km/h" is reporting a scalar (wind speed). The same bureau's vector arrow on the synoptic chart, "90 km/h from the south-west", is what tells emergency services which towns are downwind. Same magnitude; the direction is what makes it actionable.

A scalar has magnitude only (distance, speed, time, mass). A vector has magnitude AND direction (displacement, velocity, acceleration, force). A vector's magnitude is always positive; the direction is separate, essential information.

Pause, copy the two definitions and the two example lists (scalars vs vectors) into your book before moving on.

Three of these are scalars. Which one is the odd one out (a vector)?

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Why direction changes everything

02
Why Direction Changes Everything
+5 XP

We just saw that vectors carry a direction while scalars do not. That raises a question: does that direction actually change any answer, or is it just extra labelling? This card answers it → with the drone trip, where the scalar and the vector give genuinely different results.

Run the drone trip again. It flies 6 km out to a remote clinic, drops the samples, and flies 6 km back. Ask "how far did it travel?" and the answer is a scalar: 12 km of path. Ask "how far is it from base now?" and the answer is a vector: it is back where it started, so 0 km.

The path length (a distance, scalar) added up to 12 km because distance never decreases, every metre of travel counts. But the change in position (a displacement, vector) is zero, because the outbound trip east and the return trip west cancel. Cancellation is only possible because the two legs point in opposite directions, and only vectors carry direction.

The key idea: scalars accumulate; vectors can cancel. Whenever a quantity can "undo itself" by reversing direction, you are dealing with a vector. Whenever a quantity only ever piles up, you are dealing with a scalar.
Exam trap: students write "displacement = 12 km" for an out-and-back trip by quietly using the distance. Displacement is measured from start to finish only the path in between is irrelevant.

Distance (scalar) is the total path length and only ever grows. Displacement (vector) is the change in position from start to finish and can be zero or negative because opposite directions cancel. Same trip, two different numbers.

Pause, copy the drone example and the rule "scalars accumulate, vectors can cancel" into your book before moving on.

True or false: a runner completes one full lap of a 400 m track and returns to the start. Their distance is 400 m but their displacement is 0 m.

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Draw vectors to scale

03
Drawing Vectors, Arrows to Scale
+5 XP

We just saw that direction lets vectors cancel where scalars cannot. That raises a question: how do we record a vector on paper so its direction is unmistakable? This card answers it → we draw it as an arrow, where the length is the magnitude and the arrowhead is the direction.

A number alone cannot show "east" or "30° above the road". So we draw vectors as arrows. Everything about the arrow means something: how long you draw it, and which way it points.

An arrow representing a vector has two carefully controlled features:

  • Length = magnitude. You pick a scale (e.g. 1 cm represents 10 m/s) and draw the arrow that many times long. A 30 m/s velocity is drawn three times longer than a 10 m/s velocity.
  • Arrowhead = direction. The way the head points is the direction of the vector. The tail is the starting point; the head is where it points.

Notation: writing a vector down

In handwriting a vector symbol is written with an arrow above it, $\vec{v}$, or in bold in print, $\mathbf{v}$. Its magnitude (just the size) is written without the arrow, $v$ or $|\vec{v}|$. So $\vec{v}$ might be "20 m/s north" while $v = 20$ m/s is only the speed.

You always state a vector as magnitude + direction: "12 m east", "9.8 m/s² downwards", "40 N at 30° above horizontal".

Equal and negative vectors: two vectors are equal if they have the same magnitude and the same direction, it does not matter where on the page they are drawn. A negative vector $-\vec{a}$ has the same length as $\vec{a}$ but points the exact opposite way.

A vector is drawn as an arrow: length = magnitude (to a chosen scale), arrowhead = direction. Symbol $\vec{v}$ (or bold $\mathbf{v}$); magnitude is $v$ or $|\vec{v}|$. Equal vectors share magnitude AND direction; the negative vector $-\vec{a}$ has equal length but opposite direction.

Pause, copy the arrow rule (length = magnitude, head = direction), the notation, and the equal/negative vector definitions into your book before moving on.

Match each vector term to its meaning.

  • Magnitude
  • Direction
  • Equal vectors
  • Negative vector
  • same magnitude, opposite direction
  • same magnitude and same direction
  • shown by the way the arrowhead points
  • the size of the vector, shown by the arrow length
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Set a sign convention

04
Direction on a Line, The +/− Convention
+5 XP

We just saw that direction is drawn with an arrowhead. That raises a question: when all the motion is along a single straight line, do we really need to draw arrows every time? This card answers it → no, in one dimension we replace "east/west" with a simple + and − sign convention.

When everything happens along one line, a car on a straight road, a lift in a shaft, there are only two possible directions. Drawing arrows is overkill. Instead we agree on a sign convention: pick one direction to be positive, and the opposite direction is automatically negative.

For example, choose east = positive (+). Then west is negative (−). A displacement of "5 m east" becomes $+5$ m, and "5 m west" becomes $-5$ m. The sign now is the direction, no arrow needed. (You may pick either direction as positive; you just have to state your choice and stick to it.)

One-dimensional vector notation $+x$  means  magnitude $x$ in the chosen positive direction  ·  $-x$  means  magnitude $x$ in the opposite direction
Why this matters: the +/− convention is what lets us add and subtract vectors with ordinary arithmetic instead of geometry, but only when they lie on the same line. It powers every motion-graph and equation-of-motion calculation later in this module.

In one dimension, choose one direction as positive (+); the opposite is negative (−). The sign then carries the direction: +5 m and −5 m are equal magnitudes pointing opposite ways. State your sign convention before you calculate.

Pause, copy the sign-convention rule and one worked example (e.g. east = +, so 5 m west = −5 m) into your book before moving on.

A student sets "up = positive". An object moves 3 m downward. Complete the sentence.

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Add vectors along a line

05
Adding and Subtracting Vectors Along a Line
+5 XP

We just saw that signs replace arrows in one dimension. That raises a question: how do we combine two vectors, like two legs of a journey, once they each carry a sign? This card answers it → we simply add the signed numbers; the sign of the result gives the resultant's direction.

Two vectors on the same line are combined by adding their signed values. The single vector that results is called the resultant. Its sign tells you the direction; its size tells you the magnitude.

Take the convention east = +. A person walks 8 m east ($+8$ m) then 3 m west ($-3$ m). The resultant displacement is:

Adding vectors in one dimension $\vec{s} = (+8\ \text{m}) + (-3\ \text{m}) = +5\ \text{m}$  →  5 m east

Subtracting a vector is the same as adding its negative: $\vec{a} - \vec{b} = \vec{a} + (-\vec{b})$. This is exactly how we will later find a change in velocity (for acceleration): subtract the start vector from the end vector.

Worked example, combining two legs
  1. State the convention. Let east = positive (+), so west = negative (−).
  2. Write each vector with its sign. 12 m east = $+12$ m; 20 m west = $-20$ m.
  3. Add the signed values. $\vec{s} = (+12) + (-20) = -8$ m.
  4. Interpret the sign. The result is negative, so the resultant displacement is 8 m west. (Distance travelled, a scalar, would be $12 + 20 = 32$ m, different number.)
Exam trap: never drop the sign on the final line. "$\vec{s} = 8$ m" is incomplete, you must report "$-8$ m" or "8 m west". The magnitude alone hides the direction the marker is looking for.

To add vectors on a line, give each a sign from your convention, then add the signed numbers. The resultant's sign is its direction; its size is its magnitude. Subtracting a vector = adding its negative: $\vec{a}-\vec{b}=\vec{a}+(-\vec{b})$. Always keep the sign on the final answer.

Pause, copy the four-step method and the worked example ($+12 + (-20) = -8$ m = 8 m west) into your book before moving on.

Taking north as positive, a cyclist rides 15 m north then 9 m south. What is the resultant displacement?

Scalar versus vector representation A scalar shown as a number alone (speed equals 30 metres per second) compared with a vector drawn as an arrow whose length is the magnitude and whose arrowhead is the direction, plus a negative vector of equal length pointing the opposite way. SCALAR magnitude only speed = 30 m/s no direction needed A single number + unit fully describes it. VECTOR magnitude + direction v = 30 m/s east length = magnitude · head = direction −v = 30 m/s west same length, opposite direction
A scalar is a number alone; a vector is an arrow, length shows magnitude, the head shows direction. The lower arrow is the negative vector: equal length, opposite way.
Interactive Tool, Vector Builder Open fullscreen ↗

What to do: drag the arrow to change its length and direction. Watch how the magnitude (length) and direction (arrowhead) update independently, then make a negative vector by pointing it the opposite way at the same length.

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Apply it yourself

Activities

Activity 1, Sort and Justify
UnderstandBand 3

Classify each quantity below as a scalar or a vector, and for each, write one sentence justifying your choice by referring to direction.

  1. An aircraft cruising at 850 km/h to the north-east
  2. The 42.2 km length of a marathon course
  3. A lift accelerating at 1.5 m/s² upward
  4. The 3.5 s a sprinter's reaction took
  5. The 60 km/h shown on a speedometer

Two of these statements are true. One is a lie. Find the lie.

Activity 2, Add Along a Line
ApplyBand 4

Take east = positive. For each trip, write each leg with its sign, add to find the resultant displacement (with direction), and state the total distance separately.

  1. 10 m east, then 4 m east.
  2. 10 m east, then 10 m west.
  3. 6 m west, then 14 m east, then 2 m west.

For each: which answer (displacement or distance) needed the signs, and why?

✓

Show what you have learned

Quick recall, Scalars and Vectors
+5 XP

A fresh five-question set drawn from this lesson's bank, feedback shown immediately.

Pick your answer, then rate your confidence.

Multiple Choice, 5 Questions
checkpoint

1. Which of the following lists contains only scalar quantities?

  1. Distance, displacement, speed
  2. Distance, speed, time, mass
  3. Velocity, acceleration, force
  4. Displacement, velocity, temperature

2. A vector is drawn as an arrow. What does the length of the arrow represent?

  1. The direction of the vector
  2. The time the vector acts for
  3. The magnitude of the vector, to a chosen scale
  4. Whether the vector is positive or negative

3. Taking right as positive, a toy car moves 7 m right then 12 m left. Its resultant displacement is:

  1. −5 m (5 m left)
  2. +5 m (5 m right)
  3. +19 m (19 m right)
  4. −19 m (19 m left)

4. The vector $-\vec{a}$ (the negative of vector $\vec{a}$) has:

  1. A smaller magnitude than $\vec{a}$ and the same direction
  2. A magnitude of zero
  3. The same direction as $\vec{a}$ but twice the magnitude
  4. The same magnitude as $\vec{a}$ but the opposite direction

5. A hiker walks 4 km north, then 4 km south, taking 2 hours. Which statement is correct?

  1. Both the distance and the displacement are 8 km
  2. The distance is 8 km but the displacement is 0 km
  3. Both the distance and the displacement are 0 km
  4. The distance is 0 km but the displacement is 8 km
Short Answer, 9 marks
+5 XP

UnderstandBand 3(2 marks) 1. Define the terms scalar and vector, and give one example of each from the study of motion.

ApplyBand 4(3 marks) 2. Taking east as positive, a courier walks 30 m east, then 50 m west. Calculate the resultant displacement (with direction) and state the total distance travelled. Explain why the two answers differ.

AnalyseBand 5(4 marks) 3. A student claims, "A displacement of −12 m is smaller than a displacement of +5 m, because −12 is less than +5." Using the ideas of magnitude and direction, explain what is correct and what is wrong about this claim, and state the magnitude of each displacement.

Show all answers

Multiple choice

Q1, B. Distance, speed, time and mass all have magnitude only. Displacement, velocity, acceleration and force are vectors (they carry direction), so any list containing them (A, C, D) is not "scalars only".

Q2, C. In an arrow representation, the length encodes the magnitude to a chosen scale, while the arrowhead encodes the direction. Length is not the direction (that is the head) and has nothing to do with time or sign.

Q3, A. Right = +, so 7 m right = +7 m and 12 m left = −12 m. Resultant = (+7) + (−12) = −5 m, i.e. 5 m left. (The total distance, a scalar, would be 7 + 12 = 19 m, that is the distractor in C.)

Q4, D. A negative vector has the same magnitude (same length) as the original but points in the exact opposite direction. The minus sign reverses direction; it does not shrink the size or zero it out.

Q5, B. Distance is the total path length: 4 + 4 = 8 km. Displacement is the change in position from start to finish: the hiker returns to the start, so displacement = 0 km. The opposite directions cancel for the vector but not for the scalar.

Short Answer, Model Answers

Q1 (2 marks): A scalar is a quantity with magnitude (size) only and no direction, for example, distance (or speed, time, mass). A vector is a quantity with both magnitude and direction, for example, displacement (or velocity, acceleration). 1 mark for both correct definitions, 1 mark for a valid example of each.

Q2 (3 marks): Taking east as positive: leg 1 = +30 m, leg 2 = −50 m. Resultant displacement = (+30) + (−50) = −20 m, i.e. 20 m west. Total distance travelled = 30 + 50 = 80 m. The answers differ because displacement is a vector measured from start to finish, so the opposite directions partly cancel; distance is a scalar that simply adds up the whole path with no cancellation. (1 mark resultant with direction, 1 mark distance, 1 mark explanation.)

Q3 (4 marks): The sign of a one-dimensional vector represents its direction, not its size, so it is wrong to say −12 m is "smaller" than +5 m. The magnitude (size) of a vector is always positive: the magnitude of −12 m is 12 m, and the magnitude of +5 m is 5 m. By magnitude, the −12 m displacement is actually the larger of the two. What is correct is that the two displacements point in opposite directions (one negative, one positive). (1 mark: sign = direction; 1 mark: magnitude is always positive; 1 mark: magnitudes 12 m and 5 m; 1 mark: −12 m is the larger magnitude / they point opposite ways.)

✓

Retrieve, reflect and finish

Check what actually stuck
Stretch, Think Ahead
stretch

You can now add vectors along a single line. In Lesson 7 you will add vectors that are not on the same line, like a drone flying east while the wind pushes it north. Predict: if a drone heads east at 8 m/s and a 6 m/s northward wind acts on it, will its resultant speed over the ground be more or less than 8 m/s? Why can't you just add 8 and 6 to get 14?

How did your thinking change?

The delivery drone's odometer read 12 km because distance is a scalar, every metre of the out-and-back path adds up. But its displacement was 0 km because displacement is a vector: the 6 km east and 6 km west are equal magnitudes in opposite directions, so they cancel. The odometer ignored direction; the GPS did not. That is the entire scalar–vector distinction in one trip, and it is the foundation for distance vs displacement (L02), speed vs velocity (L03) and everything that follows.

Module overview Next: Distance and Displacement, Path vs Position Change →

Module overview · Physics