What do you get when you add two arrows?

How to use this simulation

1. In "Adding two vectors", the start is A = 3 N and B = 4 N with 90° between them. The resultant is R = 5.00 N, the famous 3-4-5 triangle.
2. Drag an arrow head on the canvas, or move the sliders. Flip between the triangle law and the parallelogram law: the resultant never moves.
3. Sweep the angle between the vectors from 0° to 180° and watch R shrink. Compare it with "Sizes added" to see why you cannot just add magnitudes.
4. In "Boat crossing a river", turn the heading slider. Read the drift, the crossing time and the resultant speed, then switch on the shortest-path heading.
5. In "Rain and umbrella", raise the walking speed. Check that the tilt that keeps you dry matches tan θ = walking speed ÷ rain speed.

Why a boat never goes straight

Stand on one bank and aim a rowing boat at the landing directly opposite. What happens? You do not land there. You come ashore some way downstream. Every boatman knows this, so he starts out aimed a little upstream.

The rain does the same trick. It falls straight down, yet when you run or cycle it seems to charge at you from the front. Hold the umbrella straight up and your face gets soaked.

The two stories share one idea: two separate motions or forces act at once, and they point in different directions. Then you cannot just add the numbers. 3 plus 4 is not always 7. Sometimes it is 5, sometimes 1, sometimes 6.08. This lesson explains why.

From zero: scalars and vectors

Some quantities need only a number and a unit. "My mass is 50 kg." "The room is 30 °C." "The road is 2 km long." These are scalars. Adding 2 kg and 3 kg always gives 5 kg.

Others need a direction too. "I walked 5 km" is an unfinished report, because you could be anywhere on a circle. "5 km north" pins the spot. Velocity, force, displacement and acceleration are like that. They are vectors.

We draw a vector as an arrow. The length shows the size, the head shows the direction. Textbooks print vectors in bold or with an arrow overhead; here we write plain A, B and R and use |A| when we mean the size alone.

Words you must know

These terms come back in nearly every question on the topic. Learn each with its meaning.

The triangle law: head to tail

Walk 3 m east, turn, then walk 4 m north. How far are you from the start, and in which direction? Draw the first arrow, put the tail of the second on its head, and draw a third arrow from the start to the final head. That third arrow is the resultant.

This is the triangle law. If two vectors are drawn as two sides of a triangle taken in the same order, the third side, taken in the opposite order, gives their resultant in size and direction.

In the simulation default, A and B make 90° with each other, so the triangle is right-angled: 3, 4 and a hypotenuse of 5.00. Pythagoras agrees.

Does it matter which arrow goes first? No. Vector addition does not care about order: A + B = B + A. Swap the arrows in your head, or in the simulation, and the resultant ends at the same point.

The parallelogram law: two tails, one point

Two people drag a heavy sack, each with a rope tied to the same knot. The two pulls are forces with their tails at one point. Drawing one arrow after the other feels awkward here, so we use the parallelogram law.

Draw A and B from the same point. Complete the parallelogram: from the head of A, a line parallel to B; from the head of B, a line parallel to A. They meet at a corner, and the diagonal from the start to that corner is the resultant.

Opposite sides of a parallelogram are equal. So the copy of B drawn from the head of A is the triangle law in disguise. The two laws are one picture drawn two ways. In the simulation, swap laws and watch R stay put.

Size and direction of the resultant

Pictures give a good estimate. For numbers, put the axis along A. Split B into two parts: B cosθ along A and B sinθ at right angles to it. The resultant then has A + B cosθ along A and B sinθ across.

Those two parts are the legs of a right triangle, and R is its hypotenuse. So R² = (A + B cosθ)² + (B sinθ)². Expand, use sin²θ + cos²θ = 1, and the clutter cancels, leaving the famous formula.

For direction, use the tangent of the right triangle: opposite over adjacent. α is the angle between R and A. The Greek letter is only a name.

How the resultant changes with the angle

Take 3 N and 4 N and open θ. The numbers in the table are computed in code; set 3 and 4 in the simulation and turn the angle slider to check.

Two extremes are worth remembering. At θ = 0° the arrows point the same way and R is largest, A + B. At θ = 180° they oppose and R is smallest, the difference. Every other angle gives something in between. By default 3 + 4 = 7, yet R is only 5.00, because the angle is 90°.

• θ = 0°: same direction, the sizes simply add.
• θ = 90°: right angle, R = √(A² + B²).
• θ = 180°: opposite directions, the sizes subtract.
• A = B: R = 2A cos(θ/2) and the resultant bisects the angle.

Components: breaking one vector in two

The reverse job exists too. A slanting vector can be split along two axes. When you pull a cart up a slope, part of your pull moves it forward and part tries to lift it. Those parts are the components.

For a force F at angle θ to the x axis, the x component is F cosθ and the y component is F sinθ. They are at right angles, so Pythagoras brings back F.

With "Show components of R" ticked, green lines mark Rx and Ry. By default Rx = 1.45 N and Ry = 4.78 N. Add the components of A and B separately and you land on the same numbers. This is how hard problems turn easy: split every vector, add x to x and y to y, and use Pythagoras once at the end.

A boat in a river: two velocities at once

A boat can move at v through still water. But the water itself moves at u. Seen from the bank, the boat’s real velocity is the vector sum of the two.

Aim straight across and the boat’s velocity is at right angles to the current, so the resultant is √(v² + u²). The crossing time depends only on the across-the-river part of the velocity; the current adds nothing to it. It only carries the boat downstream. With a 100 m river, v = 5 m/s and u = 3 m/s, the crossing takes 20.0 s and the drift is 60.0 m.

Now there are two questions with two different answers. For the least time, aim straight across. For the least distance, meaning a landing exactly opposite, aim upstream.

The condition is zero net speed downstream: the upstream part of the boat velocity cancels the current, v sinφ = u, so sinφ = u ÷ v. If u is at least v, the right-hand side reaches 1 or more and there is no solution. A current faster than the boat makes an opposite landing impossible, however you steer.

Note that the shortest time and the shortest path are not the same thing. Taking the shortest path costs time.

Rain and umbrella: relative velocity

Rain falls straight down at v. You walk forward at u. What velocity does the rain seem to have to you? The velocity of one thing seen from another is its relative velocity. The rule: take the velocity of what you are watching and subtract the velocity of the watcher.

So rain relative to you = rain velocity − your velocity. Subtracting means adding −u, an arrow pointing backwards. Add a backward arrow to the downward one and the rain comes at you slanted.

Hold the umbrella toward where the rain comes from: from the vertical, tilted forward by θ with tan θ = your speed ÷ the rain’s speed. In the default, 4 m/s rain and a 3 m/s walk give tan θ = 0.75, θ = 36.9° and a relative speed of 5.00 m/s.

A curious point: walk faster and the rain looks both faster and more slanted to you, though in the sky nothing changed. Tick "See it from the walker" and the drops really lean.

Try it in the simulation

Each experiment takes a minute. Do them once and the formulas stay with you.

• Experiment 1: in the first scene, turn B slowly until θ = 0°. Is R the sum of the sizes? Now open θ to 180°. By how much did R fall?
• Experiment 2: make A = B and set θ to 120°. What is R? Why does it equal A?
• Experiment 3: choose the parallelogram law and check that the diagonal lies on R. Then switch to the triangle law.
• Experiment 4: keep the components on, rotate A, and find when Rx or Ry changes sign.
• Experiment 5: in the river scene set the current to 0 and swing the heading. What is the drift? Raise the current and steer again.
• Experiment 6: set the boat to 3 m/s and the current to 4 m/s. Why does the shortest-path heading show a dash?
• Experiment 7: in the rain scene untick the automatic tilt. Slide the umbrella until the label says Dry, write the angle down, and check it against tan θ.

Solved problems

Every number is computed in code. Try each on paper first, then read the steps.

Mistakes that cost marks

The same slips show up in marked scripts again and again. Read the list and spot your own.

• Adding magnitudes directly. 3 N and 4 N give 7 N only when they point the same way; otherwise R is smaller.
• Mixing up θ. It is the angle between the vectors with tails at one point. The interior angle of the triangle you draw is 180° − θ.
• Writing just A in the denominator of tan α. It must be A + B cosθ.
• In river problems, dividing the width by the resultant speed. The time uses only the across-the-river component.
• Getting the sign wrong in relative velocity. The watcher’s velocity is subtracted, which means adding its reverse.
• Leaving the calculator in radian mode when a degree answer is wanted.
• Forgetting that the resultant has the units of the vectors being added, not new ones.

Where vector addition shows up

Once you can spot it, you see it everywhere. A few samples:

• Aeroplanes and wind: the pilot aims one way, the wind pushes another, and the heading must be set in advance.
• Ferries and launches: a captain crossing a busy river does this sum without a calculator.
• Tug of war: two teams, two force vectors, one resultant that decides the winner.
• A swing hung from a tree: the pull of both ropes together balances the weight.
• A curling football: the kick and the crosswind add up to the bending path.
• Cyclone tracking: wind velocity and ocean current combine into the real path.

Exam corner: grades 9 to 11

Vector questions tend to come as river, rain or force-resultant stories. Here is a model.

Quick revision

Skim these lines the night before the exam.

• A vector has size and direction; a scalar has size only.
• Triangle law: head to tail. Parallelogram law: two tails at one point. Same resultant.
• R = √(A² + B² + 2AB cosθ) and tan α = B sinθ ÷ (A + B cosθ).
• R is largest at θ = 0° and smallest at θ = 180°.
• Components: F cosθ and F sinθ. Components of the resultant are sums of components.
• River: time = width ÷ across-speed. Opposite landing needs sinφ = u ÷ v.
• Rain: tan θ = walker’s speed ÷ rain speed, and relative velocity = rain velocity − walker’s velocity.

Frequently asked questions