Cross product of two vectors
A×B has magnitude 12 and points out of the page, at right angles to both vectors.
Vectors
Drag the glowing tips, or type values. Angles are measured from the x axis, and θ is the angle from A to B. Pick a result card to see it built in the picture.
Vector A of size 6 and vector B of size 4 with 30 degrees between them. The resultant is 9.67, 11.93 degrees from A. The dot product is 20.78 and the cross product has size 12.
Show the formulas
- Resultant
- R = √(A² + B² + 2AB cos θ) = √(6² + 4² + 2 × 6 × 4 × cos 30°) = 9.67
- Direction
- tan β = B sin θ / (A + B cos θ) = 4 × sin 30° / (6 + 4 × cos 30°), so β = 11.93° from A
- Difference
- |A - B| = √(A² + B² - 2AB cos θ) = √(6² + 4² - 2 × 6 × 4 × cos 30°) = 3.23
- Dot product
- A·B = AB cos θ = 6 × 4 × cos 30° = 20.78
- Cross product
- |A×B| = AB sin θ = 6 × 4 × sin 30° = 12
- Components
- A·B = AxBx + AyBy + AzBz = 6 × 3.46 + 0 × 2 + 0 × 0 = 20.78
- Determinant
- A×B = (AyBz - AzBy) i + (AzBx - AxBz) j + (AxBy - AyBx) k = 12k
- Unit vector
- Â = A / |A| = 1i
Step by step
- Given A = 6, B = 4 and θ = 30°.
- |A×B| = AB sin θ = 6 × 4 × sin 30° = 12
- Check with the determinant. A = 6i, B = 3.46i + 2j, so A×B = (AxBy - AyBx) k = (6 × 2 - 0 × 3.46) k = 12k.
- Right-hand rule: curl your fingers from A to B and the thumb points out of the page (+z). B×A points the other way.
Why it works
- The cross product is a vector. Its magnitude is the area of the parallelogram on A and B: AB sin θ.
- Its direction is perpendicular to both A and B, chosen by the right-hand rule.
- Swapping the order reverses the direction: B×A = -(A×B). Parallel vectors give zero, because sin 0° = 0.
- For the unit vectors, i×j = k, j×k = i and k×i = j, while i×i = j×j = k×k = 0.
- Expanding in components gives A×B = (AyBz - AzBy) i + (AzBx - AxBz) j + (AxBy - AyBx) k.
Cross product of two vectors: quick answers
What is the magnitude of the cross product?
AB sin θ. For 6 and 4 at 30 degrees it is 12, which is also the area of the parallelogram on the two vectors.
Which way does A×B point?
At right angles to both vectors, by the right-hand rule: fingers from A to B, thumb along A×B. B×A points the opposite way.
When is the cross product zero?
When the vectors are parallel or opposite (θ = 0 or 180 degrees), because sin θ = 0. Torque is r×F and the magnetic force is qv×B.