Vector Addition and Resultant Calculator

Q: When should I use the cosine rule instead of components?

When the question hands you the angle between the two vectors rather than each vector's own direction — then R = √(A² + B² + 2AB cos θ) is quicker, and tan α = B sin θ/(A + B cos θ) gives the direction relative to A. For three or more vectors, or anything in three dimensions, components are the only practical method. Note the plus sign in that cosine rule: it is the triangle cosine rule with the angle replaced by its supplement, which is why it differs from the version in a maths textbook.

Two vectors in, one resultant out — magnitude and direction, with every component shown so you can follow the working. It handles both of the angle conventions that trip students up: anticlockwise from the x-axis as physics writes it, and clockwise from north as navigation does, and the triangle is drawn to your own numbers.

Last updated 11 Nov 2026

Resultant of two vectors, by magnitude and direction

Resolve, add the components, recombine — never add the magnitudes

Magnitude of the first vector AAny unit — newtons, metres, metres per second — as long as B uses the same one. The resultant comes back in that unit. A magnitude is never negative; a vector pointing the other way is the same magnitude with 180° added to its angle.

Direction of A (degrees)Measured according to the convention you pick below. Negative angles and angles over 360° are both fine and are handled correctly.

Magnitude of the second vector BSame unit as A.

Direction of B (degrees)Same convention as A.

Which angle conventionThese two run in opposite directions from different starting lines, and converting between them is θ = 90° − bearing. A bearing of 060° is 30° anticlockwise from the x-axis; a bearing of 000° (due north) is 90°. Navigation, geography and most “a ship sails on a bearing of…” questions use bearings; everything in a physics textbook uses the first one unless it says otherwise.

Your two vectors, drawn to scale and to their real directions. Blue is A, green is B placed head-to-tail on the end of A, and the thick red arrow from the origin to the far end is the resultant. The faint dashed arrows complete the parallelogram, which is the same construction seen the other way round, and the red arc measures the resultant's direction round from the x-axis.

9.2820Example

Vector A has magnitude 5.0 at 30° anticlockwise from the x-axis; vector B has magnitude 7.0 at 110°. Find the magnitude and direction of the resultant.

Components first — and the cosine rule, for when a question insists on it

Rx = A cos θ_A + B cos θ_B · Ry = A sin θ_A + B sin θ_B · R = √(Rx² + Ry²) · R = √(A² + B² + 2AB cos θ)

θ_A, θ_B: each vector’s own direction, measured from the same reference line in the same sense.
Rx, Ry: the components of the resultant: the x parts added together and the y parts added together, separately. This is OpenStax Equation 2.24.
R = √(Rx² + Ry²): the magnitude, by Pythagoras on the two sums.
θ: in the last form, the angle between the two vectors — not either vector’s own direction. NCERT Equation 3.24a. Mixing this θ up with θ_A is the single most common error on this topic.
tan α = B sin θ / (A + B cos θ): NCERT Equation 3.24f: the angle between the resultant and A. Useful when a question gives you the angle between the vectors rather than their individual bearings.

Worked example

Vector A has magnitude 5.0 at 30° anticlockwise from the x-axis; vector B has magnitude 7.0 at 110°. Find the magnitude and direction of the resultant.
Resolve A. Ax = 5.0 cos 30° = 5.0 × 0.86603 = 4.3301. Ay = 5.0 sin 30° = 5.0 × 0.50000 = 2.5000. Both positive, because 30° is in the first quadrant.
Resolve B. Bx = 7.0 cos 110° = 7.0 × (−0.34202) = −2.3941. By = 7.0 sin 110° = 7.0 × 0.93969 = 6.5778. Note the minus sign — 110° is in the second quadrant, so the x component points backwards. A calculator gives this automatically; a student drawing a triangle and taking cos 70° has to put the sign in by hand, and forgetting it is the classic slip.
Add the components separately. Rx = 4.3301 − 2.3941 = 1.9360. Ry = 2.5000 + 6.5778 = 9.0778. The x parts only ever meet x parts; the y parts only ever meet y parts.
Recombine with Pythagoras. R = √(1.9360² + 9.0778²) = √(3.7481 + 82.4064) = √86.1545 = 9.2820. Sanity check: that lies between |7 − 5| = 2 and 7 + 5 = 12, as every resultant of these two must.
Direction. θ = arctan(9.0778 / 1.9360) = 78.0° anticlockwise from the x-axis. Both components are positive so the answer is in the first quadrant and the plain arctangent is right; when Rx is negative you must add 180°, which is what the atan2 function behind this page does for you.
Cross-check with the cosine rule. The angle between the two vectors is 110° − 30° = 80°, so R = √(5² + 7² + 2×5×7×cos 80°) = √(74 + 70 × 0.17365) = √86.155 = 9.282. The same answer by a different route, and a reminder that the θ in that formula is the 80° between them, not the 30° or the 110°.

The same two vectors, A = 5 and B = 7, at different angles between them

Read the first and last rows together: two vectors of 5 and 7 can produce any resultant between 2 and 12, and nothing outside that range. That window — from |A − B| to A + B — is the quickest check there is on a vector-addition answer, and it instantly rules out the commonest wrong answer of all, which is to add the magnitudes and get 12 whatever the angle.
Angle between	Resultant R	Why
0°	12.000	Same direction — the maximum, and the only time R = A + B
30°	11.609	Still close to the sum
60°	10.440	cos 60° = ½, so R = √(74 + 35)
80°	9.282	This page’s default pair
90°	8.602	Right angles — plain Pythagoras, √(25 + 49)
120°	6.245	cos 120° = −½, so the cross term now subtracts
150°	3.400	Nearly opposed
180°	2.000	Opposite — the minimum, \|A − B\|

The two angle conventions, side by side

The conversion is θ = 90° − bearing, adding 360° if that goes negative. The two agree at north-east and nowhere else, which makes 045° a dangerous example to test yourself on. Switching the dropdown on this page re-reads both of your angles in the other convention, so you can see immediately how much difference it makes — and in most cases it makes a great deal.
Direction	Physics: anticlockwise from +x	Navigation: bearing, clockwise from north
East	0°	090°
North-east	45°	045°
North	90°	000°
North-west	135°	315°
West	180°	270°
South	270°	180°

Why magnitudes never simply add, and the method that always works

Two vectors of 5 and 7 do not make 12 unless they point the same way. They can make anything from 2 to 12, and which it is depends entirely on the angle between them. That is the one idea on this page: a vector carries a direction as well as a size, and the direction decides how much of each one reinforces the other. Two tugboats pulling a barge in slightly different directions achieve less than the sum of their pulls, and two forces of 5 N and 5 N at 120° to each other produce a resultant of exactly 5 N.

The method that never fails is to resolve. Split each vector into the part along x and the part along y, which are ordinary numbers and can be added like ordinary numbers. Add all the x parts together, add all the y parts together, and you have the resultant in component form. Then Pythagoras on those two sums gives the magnitude, and an arctangent gives the direction. The method scales to three vectors, to ten, and to three dimensions, where the triangle construction becomes impossible to draw.

Signs are where the marks go. A vector at 110° has a negative x component — it points up and to the left — and a vector at 200° has both components negative. A calculator handles this automatically if you give it the full angle, which is why this page asks for a direction rather than an angle inside a triangle. A student who draws the triangle, takes the acute angle of 70°, computes 7 cos 70° = 2.394 and writes it as positive has made the error the whole convention exists to prevent, and the resultant comes out as 11.0 instead of 9.28 — wrong, but not absurd enough to notice.

The reverse step has its own trap. Going from components back to an angle, a plain arctangent only ever returns an answer between −90° and +90°, so it cannot tell 30° from 210°: both have the same ratio of Ry to Rx. The fix is to look at the signs of Rx and Ry and decide which quadrant you are in, adding 180° when Rx is negative. This page uses the two-argument arctangent, which does exactly that, which is why its direction figure can exceed 90° where a hand-held calculator’s would not.

The second convention problem is bearings. Physics measures angles anticlockwise from the positive x-axis, which usually means east. Navigation, geography and a great many exam questions about ships and aircraft measure bearings clockwise from north. The two run in opposite senses from different starting lines, so a bearing of 060° is 30° in the physics convention, and they happen to agree only at north-east. The dropdown on this page converts for you with θ = 90° − bearing, and switching it is the fastest way to see how much difference the convention makes to a given pair of angles.

Every trigonometric function in this calculation takes radians, because the radian is the coherent SI unit of plane angle. Degrees have to be multiplied by π/180 before any sine or cosine touches them, and the failure mode is quiet: sin(30) read as radians is −0.988 rather than 0.500, and the answer that comes out is a number rather than an error. If a vector answer is wildly wrong and the working looks right, check the angle mode first. The projectile page has a table of the degree and radian values side by side.

Subtraction is addition in disguise. A − B means A + (−B), and −B is B with 180° added to its direction. That is why the difference figure appears below the sum here: relative velocity, change of velocity and change of momentum are all vector subtractions, and students who can add vectors confidently often stall on them. A car turning a corner at constant speed has a change of velocity that is not zero, because the direction changed, and working out what it is means subtracting one vector from another. The relative velocity page does that case properly.

One last check worth making every time: the resultant must lie between |A − B| and A + B. It is one line, it costs nothing, and it catches the two biggest errors at once — adding the magnitudes, and dropping a minus sign on a component. If your answer is outside that window, it is wrong, and you know it before the marker does.

What about components as the input? If a question gives you a vector as a pair of components rather than a magnitude and an angle, you have two choices. The direct one is to convert: the magnitude is √(x² + y²) and the direction is the two-argument arctangent of y over x, and then this page takes it from there. The neater one, which is worth seeing at least once, is to notice that a component pair is a pair of perpendicular vectors. A vector of (3, 4) is a vector of 3 along 0° added to a vector of 4 along 90°, and feeding those two into this page returns a magnitude of 5 at 53.13° — which is the conversion, done by the same machinery as everything else. There is no separate theory for components; resolving and adding are the same operation run in opposite directions.

The 3-4-5 example is worth keeping in your head for exactly that reason. Perpendicular vectors of 3 and 4 give 5, not 7; perpendicular vectors of 5 and 12 give 13; and perpendicular vectors of 6 and 8 give 10. Each of those is a Pythagorean triple doing the job it was invented for. If a question’s numbers look like one of them, the answer is probably meant to be exact, and a messy decimal is a sign that a component has picked up the wrong sign somewhere.

Finally, a note on what this page deliberately does not do. It adds exactly two vectors in exactly two dimensions. That is the case school questions overwhelmingly set, and keeping to it lets the diagram be drawn honestly instead of being replaced by a list of numbers. For three or more vectors, do them in pairs — add the first two here, then feed that resultant back in as A with the third vector as B. Vector addition is associative, so the order you pair them in makes no difference to the final answer, and nothing is lost by working through a long list two at a time.

Frequently asked questions

Why can’t I just add the two magnitudes?

Because that only works when both vectors point in exactly the same direction. At any other angle part of each vector is working across the other rather than with it, and the resultant is smaller than the sum. For 5 and 7 the resultant can be anything from 2 to 12; it reaches 12 only at 0° and 2 only at 180°.

When should I use the cosine rule instead of components?

When the question hands you the angle between the two vectors rather than each vector’s own direction — then R = √(A² + B² + 2AB cos θ) is quicker, and tan α = B sin θ/(A + B cos θ) gives the direction relative to A. For three or more vectors, or anything in three dimensions, components are the only practical method. Note the plus sign in that cosine rule: it is the triangle cosine rule with the angle replaced by its supplement, which is why it differs from the version in a maths textbook.

My calculator gives a negative angle. Is that wrong?

No, just a different way of writing the same direction. An angle of −45° and one of 315° point the same way. This page adds 360° to any negative result so the direction always reads between 0° and 360°, which is the form most mark schemes expect, but either is correct if you say which you mean.

What is the difference between a bearing and an angle?

A bearing is measured clockwise from north and written with three digits, like 060°. A physics angle is measured anticlockwise from the positive x-axis. The conversion is θ = 90° − bearing, adding 360° if that goes negative, so a bearing of 060° is 30° and a bearing of 000° is 90°. Use the dropdown on this page rather than converting in your head — the sense reverses as well as the zero point, and that is what catches people.

How do I subtract vectors?

Reverse the second one and add. A − B is A + (−B), where −B has the same magnitude as B and a direction 180° away. In components it is simply Ax − Bx and Ay − By, which is why the component method handles subtraction without needing a new rule. The magnitude and direction of A − B are both listed in the figures above.

Does the order matter?

No. Vector addition is commutative: A + B gives exactly the same resultant as B + A, which is what the parallelogram in the diagram shows — the two routes round it, blue-then-green and green-then-blue, end at the same point. Subtraction is not commutative, though: A − B and B − A have the same magnitude and opposite directions.

I have my vectors as components. Can I still use this?

Yes, in either of two ways. Convert each pair to a magnitude and an angle first — magnitude √(x² + y²), direction the two-argument arctangent of y over x — or, more simply, treat each component as a vector in its own right: a vector of (3, 4) is 3 along 0° plus 4 along 90°. Entering those two returns 5 at 53.13°, which is the conversion done for you.

How do I add three or more vectors?

In pairs. Add the first two here, then put that resultant in as A with the third vector as B, and so on. Vector addition is associative, so the order in which you pair them up has no effect on the final answer. If you are doing it on paper, the component method is faster for long lists: write a column of x components and a column of y components, total each column once, and recombine at the end.

Related calculators

References

NCERT, Physics: Textbook for Class XI, Part I, Chapter 3 “Motion in a Plane”, §3.4 (triangle and parallelogram laws, graphical) and §3.6 (analytical). Equation 3.24a is R = √(A² + B² + 2AB cos θ) and Equation 3.24f is tan α = B sin θ / (A + B cos θ). Read through IIT Kanpur’s SATHEE mirror.
OpenStax, University Physics Volume 1, §2.3 “Algebra of Vectors” (CC BY 4.0). Equation 2.24 adds vectors by corresponding scalar components, R_x = A_x + B_x and so on, and Equation 2.25 extends it to N vectors — the component method this page implements.
BIPM, The International System of Units (SI), 9th edition (2019, updated 2026). The newton, the pascal and the kilogram: the kilogram is a unit of mass and the newton a unit of force, and the brochure keeps them in separate tables for the reason this page keeps repeating.
NIST, CODATA Internationally Recommended 2022 Values of the Fundamental Physical Constants, ASCII table read 11 November 2026. Standard acceleration of gravity 9.806 65 m s⁻², marked (exact) rather than carrying an uncertainty; Newtonian constant of gravitation 6.674 30(15)×10⁻¹¹ m³ kg⁻¹ s⁻². physics.nist.gov/cuu/Constants/Table/allascii.txt