Vector Cross Product Calculator

Worked example

A = (3, −2, 5), B = (1, 4, −1):

• i component: A_yB_z − A_zB_y = (−2)(−1) − (5)(4) = 2 − 20 = −18
• j component: A_zB_x − A_xB_z = (5)(1) − (3)(−1) = 5 + 3 = 8
• k component: A_xB_y − A_yB_x = (3)(4) − (−2)(1) = 12 + 2 = 14

So A × B = (−18, 8, 14), with magnitude √((−18)² + 8² + 14²) = √(324 + 64 + 196) = √584 ≈ 24.166.

Why the result is perpendicular to both inputs

You can verify perpendicularity with the dot product: (A × B) · A should equal zero, since a vector can't have a component along a direction it's perpendicular to. Checking: (−18)(3) + (8)(−2) + (14)(5) = −54 − 16 + 70 = 0. The same check against B also comes out to zero — that's the defining geometric property of the cross product, not a coincidence of this example.

Order matters

Cross product is anti-commutative: B × A gives the exact negative of A × B, (18, −8, −14) in this case, not the same vector. Swapping the input order flips which direction the resulting perpendicular vector points, following the right-hand rule.

Reading the 3D plot

The canvas view is a simplified isometric projection (a fixed-angle 2D approximation of 3D space), not an interactive or physically accurate 3D renderer — it's meant to give a rough sense of how vector A, vector B, and their cross product relate spatially, not to support precise visual measurement.