Scalars & Vectors
Some quantities only need a number. Others are useless until you say which way. This lesson takes you from "is it a vector?" all the way to solving a three-vector resultant the way the exam wants it — with things you can drag, slide and check yourself on.
Use ← → arrow keys to move between sections. Your progress saves automatically on this computer. Bring a scientific calculator — every lab shows the same numbers you should be getting.
Scalar vs vector
Every physical quantity you will meet this year falls into one of two camps. The test is a single question: does the direction change the answer? If saying "5 metres east" is different from saying "5 metres west", you are holding a vector.
Scalar quantities
Fully described by a magnitude — a number and a unit. Nothing else is needed. They add like ordinary arithmetic: 3 kg + 4 kg = 7 kg, always.
- Distance (25 m)
- Speed (60 km/h)
- Mass (52 kg)
- Time (9.8 s)
- Temperature (32 °C)
- Work, energy, power
Vector quantities
Need magnitude AND direction. They do not add like ordinary numbers: 3 N + 4 N can give anything from 1 N to 7 N depending on the angle between them.
- Displacement (25 m, east)
- Velocity (60 km/h, north)
- Acceleration (9.8 m/s², down)
- Force and weight (510 N, down)
- Momentum, impulse
- Electric and magnetic fields
Remember — the pairs that trap people
Distance (scalar) vs displacement (vector). Speed (scalar) vs velocity (vector). Mass (scalar, kg) vs weight (vector, N, always pointing down). Examiners love these three pairs — learn them by heart.
Side by side
| Quantity | SI unit | Type | Why |
|---|---|---|---|
| Distance | m | Scalar | Total ground covered. "I walked 2 km" is complete on its own. |
| Displacement | m | Vector | Straight line from start to end — "2 km, north-east". |
| Speed | m/s | Scalar | What the speedometer reads. It never shows a direction. |
| Velocity | m/s | Vector | Speed plus heading. Turning changes velocity even at constant speed. |
| Mass | kg | Scalar | How much matter. The same anywhere in the universe. |
| Weight | N | Vector | The pull of gravity — a force, so it points toward the earth's centre. |
| Work / Energy | J | Scalar | Comes from a dot product — direction cancels out of the answer. |
| Momentum | kg·m/s | Vector | p = mv. A vector times a scalar is still a vector. |
Writing a vector down
Handwritten
An arrow over the letter: A. This is what you write on paper and on the board.
Printed
Textbooks use bold instead: A. Same meaning, easier to typeset.
Magnitude only
Bars or a plain letter: |A| or A. This part is a scalar and is never negative.
Describing a vector
A vector is drawn as an arrow: its length is the magnitude, its tip shows the direction. Where it sits on the page does not matter — slide it anywhere and it is still the same vector, as long as the length and the heading do not change.
Magnitude
How long the arrow is, using a scale like 1 cm : 1 m. Always positive.
Direction
Where the tip points — an angle, measured from an agreed reference line.
Tail & head
The tail is where it starts, the head (arrowhead) is where it ends.
Three ways to state the same direction
| Convention | Looks like | Read it as |
|---|---|---|
| Standard position | 60° | 60° counterclockwise from the +x axis. Use this one in formulas. |
| Compass bearing | N 30° E | Face north, then swing 30° toward east. Same direction as 60° above. |
| From the horizontal | 60° above the horizontal | How problems usually phrase a ramp, a rope or a launch angle. |
Remember
Convert everything to standard position before you touch sin or cos. East = 0°, North = 90°, West = 180°, South = 270°. Mixing a compass bearing into a component formula is the single most common way to lose marks in this topic.
Try it: drag the vector
Drag the tip anywhere on the plane. Watch the magnitude, the angle and the two components change together — they are four views of the same arrow. Click the canvas and use the arrow keys for fine control.
Vectors that are related
Equal vectors
Same magnitude, same direction — even if they are drawn in different places.
Negative vector
−A has the same length but points 180° the other way.
Scaled vector
3A is three times as long, same heading. −2A is twice as long, reversed.
Breaking a vector into components
Any slanted vector can be replaced by two perpendicular ones: a piece along x and a piece along y. Those are its components. Together they have exactly the same effect as the original — and unlike the original, they are easy to add.
θ measured counterclockwise from the +x axis · cosine → x, sine → y
Vector → components
Ax = A cos θ
Ay = A sin θ
Use this when a problem gives you a magnitude and an angle.
Components → vector
A = √(Ax² + Ay²)
θ = tan⁻¹ (Ay ÷ Ax), then fix the quadrant
Use this at the end, to state the resultant.
Try it: the component resolver
Set a magnitude and an angle, then watch the shaded right triangle. The vector is always the hypotenuse; the components are the two legs. Notice what happens to the signs as the angle passes 90°, 180° and 270°.
The sign is part of the answer
A component can be negative — that is how the maths remembers "leftward" or "downward". Dropping a minus sign turns a vector pointing south-west into one pointing north-east. Never write a component as an absolute value.
Angles worth memorising
| θ | cos θ | sin θ | What the vector is doing |
|---|---|---|---|
| 0° | 1 | 0 | All x, no y — pointing due east. |
| 30° | 0.866 | 0.500 | Mostly horizontal. |
| 45° | 0.707 | 0.707 | Equal components — perfectly diagonal. |
| 60° | 0.500 | 0.866 | Mostly vertical. |
| 90° | 0 | 1 | All y, no x — pointing due north. |
| 180° | −1 | 0 | Due west: Ax negative, Ay zero. |
| 270° | 0 | −1 | Due south: Ax zero, Ay negative. |
Adding vectors graphically
The resultant is the single vector that could replace all the others. Drawn to scale, you can find it with a ruler and a protractor — and seeing it drawn is what makes the algebra later make sense.
Head-to-tail (polygon)
Draw the first vector. Start the second one at the head of the first. Keep going. The resultant runs from the tail of the first to the head of the last. Works for any number of vectors.
Parallelogram
Draw both vectors from the same origin. Complete the parallelogram with two dashed sides. The diagonal from that origin is the resultant. Handles two vectors at a time.
Try it: the vector adder
Head-to-tail: draw A, start B where A ended, start C where B ended. The resultant runs from the very first tail to the very last head — and the order does not matter.
Order does not matter
A + B = B + A. Press Swap A and B — the path changes shape but the resultant lands on exactly the same point.
Subtracting
A − B = A + (−B). Reverse B by adding 180° to its angle, then add it normally. There is no separate rule to learn.
Closed polygon = zero
If the head of the last vector lands back on the first tail, the resultant is zero and the system is in equilibrium.
Remember — how big can a resultant get?
For two vectors of size 5 and 3: the largest resultant is 8 (same direction), the smallest is 2 (opposite directions), and anything in between is possible. If your answer falls outside that range, you have made an arithmetic mistake.
The component method
Rulers and protractors are approximate; the component method is exact. It is also the method every exam expects. Four steps, every single time — no matter how many vectors you are given.
Resolve
Break every vector into Ax = A cos θ and Ay = A sin θ. Keep the signs.
Add
Rx = ΣAx and Ry = ΣAy. Add x's with x's, y's with y's — never mix.
Magnitude
R = √(Rx² + Ry²). Pythagoras on the two totals.
Direction
α = tan⁻¹|Ry ÷ Rx|, then correct it for the quadrant.
Step 4 is where marks are lost: the quadrant
Your calculator only ever returns an angle between −90° and +90°, so it cannot tell Quadrant II from Quadrant IV. You decide, by looking at the signs of Rx and Ry. Tap a quadrant to see its rule.
Try it: the component-method solver
Enter each vector's magnitude and direction. The panel writes out the full solution in the order your teacher wants to see it — copy the structure, not just the answer.
Quadrant corrections at a glance
| Quadrant | θ range | Signs | Correction |
|---|
Worked example
Problem: A hiker walks 5.0 km at 30° north of east, then 3.0 km at 200°. Find the resultant displacement.
STEP 1 — Resolve Ax = 5.0 cos 30° = +4.330 km Ay = 5.0 sin 30° = +2.500 km Bx = 3.0 cos 200° = -2.819 km By = 3.0 sin 200° = -1.026 km STEP 2 — Add like with like Rx = 4.330 + (-2.819) = +1.511 km Ry = 2.500 + (-1.026) = +1.474 km STEP 3 — Magnitude R = √(1.511² + 1.474²) = √(2.283 + 2.173) = 2.11 km STEP 4 — Direction α = tan⁻¹|1.474 ÷ 1.511| = 44.3° Rx is positive, Ry is positive → Quadrant I → θ = α ANSWER: R = 2.11 km at 44.3° (N 45.7° E)
Sanity check: the second leg points back toward the south-west, so the resultant should be much shorter than 5.0 + 3.0 = 8.0 km. It is. Always ask the same question at the end: is this answer reasonable?
Distance vs displacement — vectors in real life
This is the pair that shows up in every quarterly exam, and it is also the one you use without noticing every time you open a map app. Tap each scene to reveal the physics at work.
Distance
The total path length actually travelled. It only ever grows — it can never decrease and is never negative. Symbol d, unit metres.
Displacement
The straight arrow from the starting point to the ending point, with a direction. It can shrink, and it is zero for any round trip. Symbol d or Δx.
Remember — the rule that is always true
|displacement| ≤ distance. They are equal only when the motion is in a straight line without turning back. Any curve, any detour, any U-turn makes the distance bigger.
Vectors around you
Same trip, two answers
| The trip | Distance | Displacement |
|---|---|---|
| Walk 3 m east, then 4 m north | 7 m | 5 m at 53.1° (N 36.9° E) |
| Walk 5 m east, then 5 m west | 10 m | 0 |
| One full lap of a 400 m oval | 400 m | 0 |
| Half a lap of the same oval (r ≈ 63.7 m) | 200 m | ≈127 m, across the field |
| Drive 12 km north on a straight highway | 12 km | 12 km north |
Find what's missing
So far every question has handed you everything and asked for the resultant. Real problems are rarely that polite. They hand you some of it and ask for the rest — and the method never changes, only the rearranging does. Three steps here: one vector, then the words, then the equation.
Part 1 — one vector, one missing piece
A single vector has four parts: its magnitude A, its direction θ, and its two components Ax and Ay. They are not four independent facts. Any two of them fix the other two — so the only question is which two you were handed.
Tap the two parts the problem gives you.
Why one pairing gives you two answers
Give it A and Ax and it will hand you two vectors. That is not a bug in the maths — Pythagoras only ever tells you the size of the other component, never its sign, so the vector could be above the axis or below it and every number you were given still fits. The words of the problem are what break the tie: "north of east", "below the horizontal", a sketch. If nothing does, the honest answer is both.
Part 2 — turning the words into symbols
This is the step that actually loses marks. The arithmetic is four steps you have already practised; the hard part is reading a sentence and deciding what is A, what is R, and what is being asked for. So do that on its own, before touching a calculator. Some of the values below are not in the problem at all — they are the tempting misreadings.
Tap a value, then tap the slot it belongs in. Tap a filled slot to send it back.
What is the question asking you to find?
Part 3 — the missing vector in an equation
Now the equation itself. Two vectors and their resultant are tied together by A + B = R; give any two and the third is fixed.
Both legs known
"Walk 4 km east, then 3 km north. Find the resultant." Add the components: R = A + B.
One leg missing
"You must end up 150 N at 40°, and rope A gives 120 N east. What does rope B give?" Subtract: B = R − A.
The first leg missing
"The total is 90 m at 65°; the second leg was 50 m at 120°. What was the first?" Same move: A = R − B.
The only new idea on this page
A vector equation is two ordinary equations wearing one coat — one for x, one for y. So you never rearrange arrows. You rearrange the equation on paper, resolve everything into components, and then it is plain arithmetic twice over. Step 3 and step 4 are exactly the ones you already know.
Tap whichever vector the problem is asking for. The other two become the inputs, and the missing one is drawn as a dashed arrow — because the two you know already pin down both of its ends. Estimate it off the plane first, then compute it. If the estimate and the answer disagree, one of them is wrong. Switch to Check my answer and it will mark your working instead of doing it for you.
Worked example: the equilibrant
Problem: A force of 80 N acts at 150°. What second force must be applied so that the two forces add to zero and nothing moves?
THE EQUATION A + B = R, with R = 0 Rearranged: B = R - A = -A STEP 1 - Resolve what you were given Ax = 80 cos 150° = -69.282 N Ay = 80 sin 150° = +40.000 N Rx = 0 Ry = 0 STEP 2 - Solve component by component Bx = Rx - Ax = 0 - (-69.282) = +69.282 N By = Ry - Ay = 0 - (+40.000) = -40.000 N STEP 3 - Magnitude B = √(69.282² + 40.000²) = √(4800 + 1600) = 80.00 N STEP 4 - Direction α = tan⁻¹|-40.000 ÷ 69.282| = 30.0° Bx is positive and By is negative → Quadrant IV → θ = 360° - 30.0° ANSWER: B = 80.00 N at 330° (S 60.0° E)
Read the answer, do not just write it. 80 N at 330° is the same size as the original force and pointing exactly 180° away from it. That is what an equilibrant always is — and it is a free check on every "make it balance" question you will ever get.
The same question, wearing different clothes
| How the exam words it | What is missing | Rearrange to |
|---|---|---|
| "Find the resultant of the two forces." | R | R = A + B |
| "What third force keeps the object in equilibrium?" | B, with R = 0 | B = −A |
| "What heading must the pilot hold to fly due north?" | A (air velocity) | A = R − B |
| "The hiker ended 6 km at 20°. Leg 1 was 4 km east. Find leg 2." | B | B = R − A |
| "How much more force is still needed?" | B | B = R − A |
The mistake this page exists to kill
150 − 120 = 30, so rope B pulls 30 N. No. Magnitudes only subtract like that when the two vectors lie on the same line. Rope B in that problem pulls 96.6 N — more than three times the guess — because it also has to swing the pull round to 40°. Subtract components, never magnitudes.
Sort it: scalar or vector?
Twelve quantities. For each one ask the single question: would the answer change if I reversed the direction? Drag it into a bin, or tap the token and then tap a bin.
The reversal question
"Is −5 different from +5 here?" For mass, no — there is no such thing as negative mass in this course. For velocity, absolutely: +5 m/s and −5 m/s are moving opposite ways. That difference is the whole idea of a vector.
Scalars in disguise
Work and energy come from vectors (force and displacement) but the product strips the direction away — they are scalars. And temperature has no direction even though it has a sign.
Flashcard review
Twelve terms you need before the quiz. Tap a card to flip it. Cover the answers and say the definition out loud first — that is what makes it stick.
Practice quiz
Twelve questions, a mix of concept and computation. You get an explanation after every answer — read it even when you get it right.
The questions shuffle
Every attempt reorders the items, so you cannot memorise the sequence — only the physics.
Solve on paper first
For the computational items, work it out with your calculator before you tap. Recognising an answer is not the same as producing one.
Aim for 10 / 12
Below that, revisit the component method and the flashcards, then try again.
The whole lesson on one page
If you remember nothing else, remember these. Hit the printer icon at the top to save this page as a reviewer.
Direction is data
A scalar is finished at the number. A vector is only half-stated until you give the direction — and the direction changes the arithmetic, not just the story.
Any vector is two vectors
Ax = A cos θ, Ay = A sin θ. Splitting a slanted vector into perpendicular pieces turns geometry into arithmetic. This is the trick the whole quarter runs on.
Add components, not magnitudes
Rx = ΣAx, Ry = ΣAy, then R = √(Rx² + Ry²). Adding magnitudes directly is only legal when the vectors point the same way.
The calculator cannot see the quadrant
tan⁻¹ returns −90° to +90°. Check the signs of Rx and Ry, name the quadrant, then apply the correction. Half the lost marks in this topic live here.
One-page cheat sheet
| What you need | Formula | Remember |
|---|
Before you close this
On exam day, write these three lines at the top of your scratch paper before you read a single item: Ax = A cos θ · Ay = A sin θ, R = √(Rx² + Ry²), and the quadrant rules (I: α · II: 180−α · III: 180+α · IV: 360−α). Most of the vector items fall out of those three lines.