Cross Product Calculator

The cross product of two three-dimensional vectors, vector A = (a_x, a_y, a_z) and vector B = (b_x, b_y, b_z), results in a third vector C = (c_x, c_y, c_z) that is orthogonal (perpendicular) to both original vectors. The mathematical formulas used to derive each component are: c_x = (a_y * b_z) - (a_z * b_y), c_y = (a_z * b_x) - (a_x * b_z), and c_z = (a_x * b_y) - (a_y * b_x). These equations stem from evaluating the determinant of a 3x3 matrix containing the standard unit vectors and the components of vectors A and B.

Imagine you have two vectors: Vector A = (2, 3, 4) and Vector B = (5, 6, 7). To find their cross product, we apply our component formulas step-by-step. First, for the x-component: c_x = (3 * 7) - (4 * 6) = 21 - 24 = -3. Next, for the y-component: c_y = (4 * 5) - (2 * 7) = 20 - 14 = 6. Finally, for the z-component: c_z = (2 * 6) - (3 * 5) = 12 - 15 = -3. Thus, the resulting cross product vector C is (-3, 6, -3).

Always double-check the coordinate system orientation (右手系 or right-hand rule) when interpreting your cross product results, as reversing the order of input vectors (A x B versus B x A) will yield an inverted vector. Additionally, remember that if the cross product of two non-zero vectors equals the zero vector, the vectors are strictly parallel.