Dot and cross products of two vectors in 2D or 3D, their magnitudes, the angle between them and the projection of one onto the other, with a drawing.
By AdminDrafted with AI writing toolsUpdated Checked examples: 5
a · b = 32, so the vectors are 12.9332° apart; the parallelogram they span has area 7.34847.
Equivalent to arccos(a·b / |a||b|), but stays accurate when the vectors are nearly parallel.
The dot product multiplies matching components and adds them, a · b = a₁b₁ + a₂b₂ + a₃b₃. The cross product of two 3D vectors is a vector perpendicular to both, and its length equals the area of the parallelogram they span. The angle between the vectors is computed as θ = atan2(|a × b|, a · b), which equals arccos(a · b / |a||b|) but stays accurate for nearly parallel vectors, and the projection of a onto b is (a · b / |b|²) b.
Physics (work and torque), 3D graphics and linear algebra are the usual uses. The default vectors a = (1, 2, 3) and b = (4, 5, 6) give a · b = 32, a × b = (−3, 6, −3) and an angle of about 12.93°.
A dot product of 0 means the vectors are perpendicular; a zero cross product means they are parallel or one is zero. In 2D mode both vectors lie in the xy-plane, so their cross product points along z: (3, 4) × (4, −3) = (0, 0, −25).
Multiply matching components and add them: a · b = a₁b₁ + a₂b₂ + a₃b₃. The result is a single number, not a vector. For a = (2, −1, 3) and b = (1, 4, −2), a · b = 2 − 4 − 6 = −8. For 2D vectors the sum has two terms instead of three.
Geometrically, a · b = |a||b| cos θ, where θ is the angle between the vectors. A positive dot product means the angle is under 90°, zero means exactly 90°, and negative means over 90°. OpenStax's dot product lesson develops both forms.
For 3D vectors, a × b = (a₂b₃ − a₃b₂, a₃b₁ − a₁b₃, a₁b₂ − a₂b₁), a vector perpendicular to both a and b. For the same a = (2, −1, 3) and b = (1, 4, −2):
So a × b = (−10, 7, 9). Check it against both inputs: (−10, 7, 9) · a = −20 − 7 + 27 = 0, and (−10, 7, 9) · b = −10 + 28 − 18 = 0.
The direction follows the right-hand rule, and swapping the order flips the sign: b × a = (10, −7, −9). OpenStax's cross product lesson covers the direction and area meaning.
| Dimensions | a | b | Dot product | Cross product |
|---|---|---|---|---|
| 3D | (1, 2, 3) | (4, 5, 6) | 32 | (−3, 6, −3) |
| 2D | (3, 4) | (4, −3) | 0 | (0, 0, −25) |
| 2D | (1, 0) | (−2, 0) | −2 | (0, 0, 0) |
| 3D | (1, 2, 3) | (2, 4, 6) | 28 | (0, 0, 0) |
A vector's magnitude is |a| = √(a₁² + a₂² + a₃²). For (1, 2, 3), |a| = √14 ≈ 3.742, and for (4, 5, 6), |b| = √77 ≈ 8.775.
The angle comes from cos θ = (a · b) ÷ (|a||b|) = 32 ÷ (√14 × √77) ≈ 0.9746, so θ ≈ 12.93°. For (2, −1, 3) and (1, 4, −2), the dot product of −8 gives an obtuse angle of about 117.81°.
Near 0° and 180° the arccos form loses accuracy in floating point, so θ = atan2(|a × b|, a · b) is the safer formula (Kahan's notes on roundoff). A zero vector has no direction, so its angle with another vector is undefined, not 0°.
The length |a × b| equals the area of the parallelogram spanned by a and b, and half of it is the area of the triangle they form. For (1, 2, 3) and (4, 5, 6), |(−3, 6, −3)| = √54 ≈ 7.348, so the triangle's area is about 3.674.
The scalar projection, the signed length of a along b, is (a · b) ÷ |b|. The vector projection is ((a · b) ÷ |b|²) b. Projecting (2, 3) onto (4, 0) gives a scalar projection of 8 ÷ 4 = 2 and the vector (2, 0).
Projection is not symmetric: projecting b onto a answers a different question. A negative scalar projection means a points partly against b.
Work is a dot product, W = F · d. A force of (3, 4) N that moves an object through (5, 0) m does 15 J of work, because only the force along the motion counts.
Torque is a cross product, τ = r × F. A 10 N force along y, applied 0.5 m from the pivot along x, gives τ = (0, 0, 5) N·m, pointing along the axis of rotation.
For matrices, use the matrix calculator; for distances and lines between points, the coordinate geometry calculator. Work and energy problems continue in the work, energy and power calculator.
Checked against: Hand calculation; exact decimal arithmetic √14, √54 and atan2(√54, 32) in degrees
Checked against: 3·4 + 4·(−3) = 0; 3·(−3) − 4·4 = −25
Checked against: b = 2a, so a × b = 0 and θ = 0
Checked against: a·b/|b| = 8/4; angle atan2(3, 2) = 56.3099324740202…°
Multiply corresponding components and add the results: a · b = a₁b₁ + a₂b₂ + a₃b₃. For (1, 2, 3) · (4, 5, 6) that is 4 + 10 + 18 = 32. The dot product also equals |a||b| cos θ, so it is positive when the angle is under 90°, 0 at exactly 90° and negative beyond it: (1, 0) · (−2, 0) = −2.
For a = (a₁, a₂, a₃) and b = (b₁, b₂, b₃), a × b = (a₂b₃ − a₃b₂, a₃b₁ − a₁b₃, a₁b₂ − a₂b₁). For (1, 2, 3) × (4, 5, 6) that gives (12 − 15, 12 − 6, 5 − 8) = (−3, 6, −3). The result is perpendicular to both vectors, follows the right-hand rule, and changes sign if the order is swapped: b × a = (3, −6, 3).
Use cos θ = (a · b)/(|a||b|). For (1, 2, 3) and (4, 5, 6), cos θ = 32/(√14 × √77) ≈ 0.974632, so θ ≈ 12.93°. Near 0° or 180° the arccos form loses accuracy, so the calculator uses the equivalent θ = atan2(|a × b|, a · b), the form William Kahan recommends in his notes on floating-point roundoff.
The two vectors are perpendicular (orthogonal), provided neither is the zero vector. (3, 4) · (4, −3) = 12 − 12 = 0, so those vectors meet at exactly 90°. In physics the same test shows that a force at right angles to the motion does no work, since work is the dot product of force and displacement.
The scalar projection of a onto b is the signed length of a along b, a · b / |b|; the vector projection is that length times the unit vector of b, (a · b / |b|²) b. Projecting (2, 3) onto (4, 0) gives a scalar projection of 8/4 = 2 and a vector projection of (2, 0). A negative scalar projection means a points partly against b.
This calculator includes 5 worked examples with independently sourced answers. They run in the test suite, and you can run them here too.