Vector Addition and Subtraction Calculator
This vector addition calculator adds and subtracts vectors, gives the resultant in exact fractions, and draws the whole thing nose to tail from your own numbers.
Type <3, 4> − <1, 2> and you get (2, 2) — along with (−2, −2), which is what you would have got had you taken them the other way round.
Adding Is the Easy Part
Components add to components. Three across plus one across is four across; four up plus two up is six up. That is all of vector addition, in every dimension.
Nothing mixes. The x parts never meet the y parts, and a z component minds its own business entirely. If a calculation ever seems to combine two different directions, something has gone wrong earlier.
Subtraction Reverses Before It Adds
A − B does not mean taking B away in some separate operation. It means turning B around and adding it: A + (−B).
Which is why the order is not a detail. <3, 4> − <1, 2> = (2, 2), but <1, 2> − <3, 4> = (−2, −2). Same length. Opposite direction. Exactly one of those answers the question in front of you.
This page prints both whenever you subtract, side by side, so the wrong one cannot be copied out by accident. It is the most common way marks are lost in this topic and it has nothing to do with arithmetic.
The Resultant, and What It Actually Is
Put the second vector so that it starts where the first one finished. Keep going for as many as you have. The resultant is the single arrow from your starting point to your finishing point.
That is why it is called the resultant: one vector with the same effect as all of them together. Two forces on a body, two legs of a journey, two velocities — the resultant is what actually happens.
The drawing under the answer is built from the same numbers as the arithmetic, not sketched separately, so the two can never disagree. If your own sketch and your own working ever part company, one of them contains a sign error and the sketch usually finds it faster.
Lengths Do Not Add Up
Here is a thing worth testing yourself on.
⟨3, 4⟩ has length 5. ⟨1, 2⟩ has length about 2.236. Their resultant ⟨4, 6⟩ has length about 7.211 — not 7.236.
Close enough to look right, and wrong. Lengths only add when the vectors point in exactly the same direction; the moment there is any angle between them, the resultant falls short. This page gives the resultant's length as an exact surd, √52 reduced to 2√13, so there is nothing to round and nothing to mistake.
| What you write | What it means | Result |
|---|---|---|
| ⟨3, 4⟩ + ⟨1, 2⟩ | add each component | (4, 6) |
| ⟨3, 4⟩ − ⟨1, 2⟩ | reverse the second, then add | (2, 2) |
| 2⟨3, 4⟩ | twice as long, same way | (6, 8) |
| −3⟨1, 2⟩ | three times as long, reversed | (−3, −6) |
Two Forces, One Answer
Almost every vector addition question outside a maths class is really a physics question wearing a different hat.
A boat points across a river at 4 m/s while the current carries it downstream at 3 m/s. Those are not two separate journeys; the boat does one thing, and the resultant ⟨3, 4⟩ describes it — 5 m/s, at an angle to the bank. Add the numbers as vectors and you get where the boat actually goes. Add the speeds as ordinary numbers and you get 7, which happens to nothing and nobody.
The same shape of question appears as forces on a beam, as two legs of a flight path, as a swimmer and a tide. A vector addition calculator is useful there precisely because the components are what the physics gives you, and the resultant is what the physics asks for.
When Everything Cancels
⟨1, 2⟩ − ⟨1, 2⟩ leaves the zero vector: no length, no direction.
Students often assume they have made a mistake. Usually they have not. In a physics question a zero resultant means the forces balanced, and that is the finding — the body is not accelerating. The drawing shows it too: the arrows walk out and come straight back to where they started.
How to Use This Vector Addition and Subtraction Calculator
Put each vector in its own brackets with a + or a − between: <3, 4> - <1, 2>. The brackets matter, because without them the signs inside a vector and the sign between two vectors look identical.
A number in front multiplies that vector, so 2<3, 4> - 3<1, 2> works in one go. Up to six vectors at a time, in two or three dimensions, and fractions stay fractions — ⟨1/2, 1/3⟩ + ⟨1/3, 1/2⟩ comes back as (5/6, 5/6) rather than (0.8333, 0.8333).
Before anything is printed the answer is checked twice over: the whole chain is added again in the opposite order and must give the same resultant, and then every vector is subtracted back off it and must leave exactly nothing. The resultant is also tested against the triangle inequality — it can never be longer than the pieces laid end to end. The length shown is worked through properly on the magnitude and direction page, and if the next line of your question needs the direction alone, the unit vector of the resultant is one step further.
Vector Addition Calculator FAQ
How do you add two vectors?
How do you subtract vectors?
Is A − B the same as B − A?
What is a resultant vector?
Can you add the magnitudes instead of the vectors?
What does a number in front of a vector do?
What if the vectors cancel out?
Can vectors with different numbers of components be added?
Add the components, reverse before you subtract, and draw it before you trust it.