Dot Product Calculator

Two forms of the same number

The dot product takes two vectors and returns a single number. There are two ways to compute it and they always agree. In components you multiply the matching parts and add them: A . B = a1 x b1 + a2 x b2 + a3 x b3. In geometry you multiply the lengths by the cosine of the angle between them: A . B = |A| x |B| x cos(theta).

OpenStax University Physics, section 2.4 calls this the scalar product, which describes exactly what comes out. The other product of two vectors, the cross product, returns a vector instead, and the two get confused in homework more often than any other pair of formulas in the subject.

Each form matches what you have in hand. Components come from a spreadsheet, a CAD file or a game engine. Magnitudes and an angle come from a physics problem, a survey or a force diagram. The calculator above accepts either and reports the angle, the projection and the work along the way.

Component form, step by step

Multiply matching components and add the products. For A = (3, 4) and B = (4, -3): 3 x 4 + 4 x -3 = 12 - 12 = 0. For A = (1, 2, 3) and B = (4, 5, 6): 1 x 4 + 2 x 5 + 3 x 6, which is 4 + 10 + 18 = 32.

The order of the vectors does not matter, because multiplication of ordinary numbers does not care about order. A 2D pair drops the third term, and a 3D pair keeps it. Getting that third term wrong is the difference between 14 and 32 in the example above, which is large enough to flip a physics answer from correct to nonsense.

Read the angle column as a check on the dot product column. A positive dot product pairs with an angle below 90, a zero pairs with exactly 90, and a negative pairs with an angle above 90. If your numbers do not follow that pattern, one of the two calculations is wrong.

The angle between two vectors

Rearranged, the geometric form gives the angle: cos(theta) = A . B / (|A| x |B|). For (1, 2, 3) and (4, 5, 6), the magnitudes are 3.7417 and 8.7750, so 32 / 32.833 = 0.9746, and the angle is 12.93 degrees. Those two vectors point in nearly the same direction, which is what a cosine close to 1 says.

Going the other way is just as useful. A force of 8 N and a displacement of 5 m at 60 degrees apart give 8 x 5 x cos(60) = 20 for the dot product, since cos(60) is 0.5. The same arithmetic appears in every work and flux calculation.

Two details keep the angle honest. The magnitudes have to be positive lengths, so a zero vector has no angle at all, and a calculator that reports one is guessing. And the cosine never exceeds 1, so a result of 1.02 means the components and the magnitudes came from different data.

Projection: how much of A points along B

The scalar projection of A onto B is A . B / |B|. It answers a question people ask constantly without naming it: if I walk in the direction of B, how far do I get in the direction A pushes me. For A = (1, 2) and B = (1, 1), the dot product is 3 and |B| is 1.4142, so the scalar projection is 3 / 1.4142 = 2.1213.

The vector projection multiplies that length by the unit vector along B: (A . B / |B|^2) x B. Here that is 3 / 2 x (1, 1), which is (1.5, 1.5). For A = (3, 4) and B = (2, 0) the scalar projection is 6 / 2 = 3 and the vector projection is (3, 0), the entire shadow of A on the x-axis.

The projection is shorter than A unless the vectors are parallel, and that inequality is the Cauchy-Schwarz bound in disguise: the dot product can never exceed the product of the magnitudes. It is also the arithmetic under a least squares fit, where the fitted values are the projection of the data onto the model space.

Work done by a force

Work is a dot product with units attached: W = F . d, in joules when the force is in newtons and the displacement in metres, which is the NIST SI definition of the joule as one newton metre. A force of (30, 40) N has a magnitude of 50 N, and pushing a crate 5 m along the x-axis, displacement (5, 0), does 30 x 5 + 40 x 0 = 150 J of work.

Only the component along the motion counts, and the dot product does that automatically. Push the same 50 N at 60 degrees to the displacement and the work is 50 x 5 x cos(60) = 125 J, not 250 J. Push at 90 degrees, as when carrying a bag while walking, and the work comes out as zero, which is why the bag's weight does not tire your arms in the physics sense no matter how far you carry it.

The sign has meaning too. A force opposed to the motion, such as friction, gives a negative dot product and negative work, and that number is the energy leaving the moving object as heat.

Cosine similarity and right-angle checks

Text search, recommendation systems and embedding models all rank items by the cosine of the angle between two vectors, which is the dot product divided by both magnitudes. Two document vectors (3, 1, 0) and (2, 2, 1) have a dot product of 8, magnitudes of 3.1623 and 3, and a cosine of 0.8433, an angle of 32.51 degrees. Dividing by the magnitudes makes the score blind to document length, which is the whole point.

At the other end of the scale sits the carpenter's check. Two vectors at right angles have a dot product of zero, so (3, 0) and (0, 4) give 0 no matter how long the walls are, and the 3-4-5 triangle people use to square a deck is the same fact with the diagonal computed from the Pythagorean theorem. A dot product of zero is the cleanest test for perpendicularity you can run on a set of coordinates.

Both uses come down to one habit: compute the dot product first, then decide whether you want to divide by the magnitudes.

Mistakes that flip the answer

Most wrong dot products come from four habits, and all four are visible in a single line of arithmetic.

The radian mistake deserves a second look because it is invisible in the formula and obvious in the answer. A calculator in radian mode reads 60 as 60 radians, which is about 9.5 full turns, and the cosine lands at -0.9524. Convert the angle once, at the top of the calculation, and the whole chain stays honest.

The last row is a scope error rather than an arithmetic one. A dot product cannot point anywhere, so any problem that asks for a perpendicular direction, a torque or a magnetic force needs the cross product, and no amount of dot product arithmetic will substitute for it.