How this instrument works
The dot product takes two vectors and returns a single number, not a third vector: A · B = x₁x₂ + y₁y₂, found by multiplying matching components and adding the results. That component recipe is not an arbitrary rule — it falls out of writing each vector in terms of the perpendicular unit vectors î and ĵ, then multiplying term by term. Because î · î = 1, ĵ · ĵ = 1, and the cross terms î · ĵ and ĵ · î both vanish (the two basis vectors are perpendicular), everything collapses to x₁x₂ + y₁y₂, with no leftover terms to account for.
Geometrically, that same number equals |A||B|cosθ, the product of the two lengths scaled by how directly they point together. That equivalence is what gives the sign its meaning: cosθ is positive whenever θ is under 90°, so a positive dot product says the vectors broadly agree in direction; a negative one says they broadly oppose, with θ past 90°; and exactly zero forces cosθ to zero, which happens only at a right angle — an algebraic test for perpendicularity that needs no protractor and no square root.
A useful edge case is dotting a vector with itself: A · A = x₁² + y₁², the sum of its squared components, which is exactly its squared length. That identity is why the dot product sits underneath distance formulas as much as angle formulas. It also generalizes cleanly where the cross product cannot: a dot product is defined for vectors of any length, from two components to a hundred, while a genuine cross product belongs specifically to three dimensions.
- Enter Vector A's components into Vector A: x and Vector A: y.
- Enter Vector B's components into Vector B: x and Vector B: y the same way.
- Read A · B — positive means the vectors broadly point together, negative means they broadly oppose.
- Watch for exactly zero: that is the signature of two perpendicular vectors, whatever their individual lengths.
- Try entering Vector B's numbers into Vector A's fields and Vector A's numbers into Vector B's fields — A · B lands on the identical figure either way.
Worked example — vectors (1, 2) and (3, 4)
Set Vector A to x1 = 1, y1 = 2 and Vector B to x2 = 3, y2 = 4. Multiply matching components and add them: A · B = (1×3) + (2×4) = 3 + 8 = 11. The sheet reports A · B as 11 — a plain positive number, with no third vector or extra component alongside it.
That 11 is positive, so the two arrows broadly point the same way: (1,2) tilts up and to the right, and (3,4) tilts further along nearly the same line, sharing an angle under 90°. Compare that to the perpendicular pair (1,0) and (0,1), whose dot product comes out to exactly 0, or to a vector paired with its own reverse, such as (2,3) and (−2,−3), which returns the strongly negative −13 — the same recipe, x₁x₂ + y₁y₂, reporting three entirely different relationships through nothing but its sign.
Questions
What does a positive or negative dot product actually mean?
A positive result means the two vectors broadly point the same way, with an angle between them under 90°. A negative result means they broadly oppose, with an angle over 90°. Neither number hands you the exact angle by itself — for that you would divide by both lengths and take the inverse cosine — but the sign alone is enough to classify two directions as roughly aligned or roughly opposed.
Why does a dot product of zero mean the vectors are perpendicular?
Because A · B equals |A||B|cosθ, and the only way that product can be zero while both vectors have nonzero length is for cosθ itself to be zero — which happens exactly at θ = 90°. Testing (1,0) against (0,1) confirms it: 1×0 + 0×1 = 0, and those two arrows are indeed at right angles, with no angle calculation required to see it.
How is the dot product different from the cross product?
The dot product returns one number, computed here as x1x2 + y1y2; a cross product instead returns a whole new vector, built from a different formula, and exists only for vectors with three components. A dot product works on vectors of any length — two numbers, three, or three hundred — while a cross product is tied specifically to three dimensions.
If I put Vector B's numbers into Vector A instead, does the result change?
No — the dot product is commutative, so A · B and B · A always agree: x1x2 + y1y2 equals x2x1 + y2y1, since ordinary multiplication does not care which factor is written first. Enter Vector B's figures into Vector A's fields and Vector A's original figures into Vector B's fields, and the reported total stays exactly where it was.
What happens when I dot a vector with itself?
A · A works out to x1² + y1², which is exactly the square of that vector's length. Dotting the vector (3, 4) with itself gives 9 + 16 = 25, and the square root of 25 recovers the familiar length 5 — the same relationship behind the Pythagorean theorem, expressed here through the dot product instead of a right triangle.
Does a zero dot product mean one of the vectors is zero?
Not necessarily — a zero result means the two vectors are perpendicular, and that includes plenty of pairs where neither vector is anywhere close to zero, such as (100, 0) and (0, 100). The dot product comes out to zero for every possible partner only when one of the inputs truly is the zero vector, (0, 0), which has no direction at all.