bùa

b×a denotes the cross product of two vectors b and a in three-dimensional Euclidean space. If a = (a1, a2, a3) and b = (b1, b2, b3), then b×a = (b2 a3 − b3 a2, b3 a1 − b1 a3, b1 a2 − b2 a1). The cross product is anti-commutative: b×a = −(a×b). The result is a vector perpendicular to both a and b, with direction given by the right-hand rule and magnitude |b×a| = |a||b| sin θ, where θ is the angle between a and b.

Geometrically, |b×a| equals the area of the parallelogram spanned by a and b, and the cross product is zero if a and b are parallel or one is the zero vector.

Key algebraic properties include bilinearity and anti-commutativity: b×(a+c) = b×a + b×c, (b1 a1 + b2 a2 + b3 a3) cross terms a bit differently expressed, and a×a = 0. More specifically, for any vectors u, v, w in R3, (u+v)×w = u×w + v×w and w×(u+v) = w×u + w×v.

Components form a convenient computation tool: b×a = (b2 a3 − b3 a2, b3 a1 − b1 a3, b1 a2 − b2 a1). The cross product is defined in three dimensions (with generalizations in higher algebra), and appears in physics and computer graphics.

Applications include torque τ = r×F, angular momentum L = r×p, magnetic force F = q v×B, and constructing normals to surfaces in 3D geometry.

Example: with a = (1,0,0) and b = (0,1,0), b×a = (0,0,−1).