The cross product is: a x b = |i j k ; $a_{1}$ $a_{2}$ $a_{3}$ ; $b_{1}$ $b_{2}$ $b_{3}$| (determinant form). Expanded: (a_{2}$$b_{3} - a_{3}$$b_{2})i - (a_{1}$$b_{3} - a_{3}$$b_{1})j + (a_{1}$$b_{2} - a_{2}$$b_{1})k. Its magnitude is |a x b| = |a||b|sin(theta). The direction is perpendicular to both a and b, following the right-hand rule. For unit vectors: i x j = k, j x k = i, k x i = j (cyclic order). Cross product is NOT commutative: a x b = -(b x a).