The cross product a x b is a vector perpendicular to both a and b, computed as the determinant of the 3x3 matrix with i, j, k in the first row. Its magnitude |a x b| = |a||b|sin(theta) equals the area of the parallelogram formed by the two vectors. The direction follows the right-hand rule. Critical properties: anti-commutative (a x b = -b x a), distributive over addition, but NOT associative. The cyclic rule gives i x j = k, j x k = i, k x i = j, with reversed order giving negative signs. Any vector crossed with itself gives the zero vector. Two non-zero vectors are parallel iff their cross product is zero. The area of a triangle with adjacent sides a and b is $\frac{1}{2}$|a x b|. The area of a quadrilateral with diagonals $d_{1}$ and $d_{2}$ is $\frac{1}{2}$|$d_{1}$ x $d_{2}$|. A unit vector perpendicular to both a and b is (a x b)/|a x b|. The cross product connects to the dot product through Lagrange's Identity.