How the cross product calculator works
The cross product is defined through a 3x3 determinant whose first row holds the unit vectors i, j, k and whose other rows hold the two input vectors. Expanding that determinant along the first row gives the three components. The calculator performs the expansion with exact fractions, displays each 2x2 minor, and then checks its own answer by dotting the result with both inputs, which must give 0. It also reports the magnitude of the result, the dot product, and the angle between the vectors, since those are the quantities most problems ask for next.
The formula
For u = (u1, u2, u3) and v = (v1, v2, v3):
u × v = (u2 v3 - u3 v2, u3 v1 - u1 v3, u1 v2 - u2 v1)
A reliable way to remember the signs is the determinant expansion i(u2 v3 - u3 v2) - j(u1 v3 - u3 v1) + k(u1 v2 - u2 v1): the middle term carries a minus sign, and the calculator's steps show it explicitly so you can see where the sign flip lands.
Worked example: (1, 2, 3) × (4, 5, 6)
The i component is 2 x 6 - 3 x 5 = 12 - 15 = -3. The j minor is 1 x 6 - 3 x 4 = 6 - 12 = -6, and with the minus sign in front the j component is +6. The k component is 1 x 5 - 2 x 4 = 5 - 8 = -3. So u × v = (-3, 6, -3). Checking perpendicularity: (-3)(1) + 6(2) + (-3)(3) = -3 + 12 - 9 = 0 and (-3)(4) + 6(5) + (-3)(6) = -12 + 30 - 18 = 0. The magnitude is sqrt(9 + 36 + 9) = sqrt(54) = 3 sqrt(6), about 7.348, which is the area of the parallelogram spanned by u and v. The dot product is 4 + 10 + 18 = 32.
Cross product versus dot product
| Property | Dot product u · v | Cross product u × v |
|---|---|---|
| Result | Scalar | Vector |
| Magnitude | |u||v| cos θ | |u||v| sin θ |
| Zero when | Vectors are perpendicular | Vectors are parallel |
| Order | u · v = v · u | u × v = -(v × u) |
| Dimensions | Any | 3D only (2D gives a scalar) |
Where the cross product shows up
| Use | Formula |
|---|---|
| Normal vector to a plane through three points P, Q, R | n = (Q - P) × (R - P) |
| Area of a triangle with sides u and v | |u × v| / 2 |
| Torque from force F at lever arm r | τ = r × F |
| Angular momentum | L = r × p |
| Volume of a parallelepiped (scalar triple product) | |u · (v × w)| |
The right-hand rule
Point the fingers of your right hand along u and curl them toward v; your thumb points along u × v. This is why the standard basis satisfies i × j = k, j × k = i, and k × i = j, and why reversing the order reverses the result. The calculator's angle output, computed from the dot product, tells you how far you have to curl: 90 degrees for perpendicular inputs, 0 or 180 for parallel ones.