Dot product calculator
(1, 2, 3) ⋅ (4, 5, 6) is 32 in the dot product calculator: 4 + 10 + 18, and the vectors are 12.93° apart.
Dot product
- a ⋅ b Multiply matching components and add. The answer is a single number, not a vector, which is why it is also called the scalar product.
- 32
- Working
- (1)(4) + (2)(5) + (3)(6) = 4 + 10 + 18 = 32
- What the sign says a ⋅ b = |a||b|cos θ, and the lengths are never negative, so the sign of the dot product is the sign of cos θ. That makes it the quickest perpendicularity test there is.
- positive: the angle is under 90°
- |a||b|cos θ The same number by the geometric definition. The two definitions agreeing is what lets you get the angle out of the components.
- √14 × √77 × cos(12.93315°) = 32
- a and b
- a = (1, 2, 3), b = (4, 5, 6)
- How a and b relate Orthogonal means the dot product is zero; parallel means the cross product is zero. Those are the two tests, and both are exact for whole-number components.
- neither parallel nor orthogonal: 12.93315° apart
| Operation | Result |
|---|---|
| a ⋅ b | 32 |
| a × b | (-3, 6, -3) |
| a + b | (5, 7, 9) |
| a - b | (-3, -3, -3) |
| |a|, |b| | √14 ≈ 3.741657, √77 ≈ 8.774964 |
| Unit vector of a | (√14/14, √14/7, 3√14/14) |
| Angle between | 12.93315° |
| Projection of a onto b | (128/77, 160/77, 192/77) |
- Every component is a whole number, so lengths, unit vectors, cosines and projections are given exactly, as fractions and simplified square roots, with decimals alongside.
- The cross product is an operation on 3D vectors. For 2D input the vectors are placed in the xy-plane, and the cross product's z component is the signed area of their parallelogram.
How the Vector Calculator works
The dot product formula is a ⋅ b = a₁b₁ + a₂b₂ + a₃b₃: multiply the matching components and add. In two dimensions drop the third term. The answer is a single number.
For a = (1, 2, 3) and b = (4, 5, 6): 1 × 4 + 2 × 5 + 3 × 6 = 4 + 10 + 18 = 32.
The geometric form is a ⋅ b = |a||b| cos θ, where θ is the angle between the vectors. Here |a| = √14 and |b| = √77, so cos θ = 32 ÷ √1078 = 0.9746 and θ = 12.93°.
The sign says how the vectors point: positive when the angle between them is under 90°, zero when they are perpendicular, and negative when it is over 90°. (1, 2) ⋅ (2, -1) = 2 - 2 = 0, so those two are at right angles.
Work in physics is a dot product, force ⋅ displacement: a 10 N force at 60° to a 5 m move does 10 × 5 × cos 60° = 25 J. A vector dotted with itself is its length squared: (1, 2, 3) ⋅ (1, 2, 3) = 14.
Vector Calculator FAQ
How is the dot product different from the cross product?
The dot product of two vectors is a number. The cross product of two 3D vectors is a third vector, perpendicular to both: (1, 2, 3) × (4, 5, 6) = (-3, 6, -3).
Does the order matter?
No: a ⋅ b = b ⋅ a, because each term is a product of two numbers.
What is the projection of a onto b?
(a ⋅ b ÷ |b|²) × b. For these vectors, 32 ÷ 77 × (4, 5, 6) = (1.662, 2.078, 2.494).