Dot Product Calculator
Enter the components of two 3D vectors to get their dot product — the single number that links the vectors to the angle between them.
Scalar product in one step
Enter the x, y and z components of vectors A and B and the calculator returns the dot product A·B = aₓbₓ + a_yb_y + a_zb_z.
Order matches
Pair each component with its partner — x with x, y with y, z with z. Negative components are fine and can make the result negative or zero.
What is the dot product?
The scalar product of two vectors
This dot product calculator finds the scalar product of two three-dimensional vectors, the single number you get by multiplying their matching components and adding the results. The dot product (also called the scalar product) takes two vectors and returns one ordinary number rather than another vector, which is why it is called "scalar". It captures how much the two vectors point in the same direction: large and positive when they align, zero when they are perpendicular, and negative when they point in opposing directions. It is the workhorse behind projections, work done by a force, lighting in computer graphics, and the angle between any two vectors.
Enter the three components of each vector to get the dot product A·B in a single step.
The dot product multiplies each pair of matching components and adds the three products together.
A·B = aₓbₓ + a_yb_y + a_zb_zEach term pairs one component of A with the matching component of B, so the x parts multiply, the y parts multiply, and the z parts multiply. Adding those three products gives a single number. Because it is a sum of products rather than another vector, the dot product is commutative: A·B equals B·A.
Suppose vector A is (1, 2, 3) and vector B is (4, 5, 6).
Multiply the x components
1 × 4 = 4 — the contribution from the x axis.
Multiply the y and z components
2 × 5 = 10 and 3 × 6 = 18 — the contributions from the y and z axes.
Add the three products
4 + 10 + 18 = 32 — the dot product A·B. The positive result tells you the two vectors point in broadly the same direction.
The dot product is more than arithmetic — its sign and size describe the geometric relationship between the two vectors. Geometrically, A·B = |A| |B| cos θ, where |A| and |B| are the lengths of the vectors and θ is the angle between them. That single identity explains every case. A positive dot product means the angle is less than 90°, so the vectors point in broadly the same direction. A negative dot product means the angle is greater than 90°, so they point in opposing directions. And a dot product of exactly zero means cos θ is zero, so θ is 90° — the two vectors are perpendicular (orthogonal) to each other. This is why the dot product is the standard test for orthogonality and the basis for projecting one vector onto another: the projection of A onto B grows directly with their dot product.
The formula is exact, but a couple of points are worth keeping in mind.
A scalar, not a vector — and watch the geometry
The dot product returns a single number, not a vector — if you need a vector perpendicular to both inputs, you want the cross product instead. The result depends on the angle through A·B = |A| |B| cos θ: a value of zero means the vectors are orthogonal (perpendicular), not that either vector is zero. This calculator works with three-dimensional vectors; pair each component with its matching partner and the same x, y and z order across both vectors.