The dot product is a scalar that is equal to the length of the projection of one vector X onto a unit vector pointing in the same direction as another vector Y, provided that X and Y start in the same location. The cross product is a vector that is perpendicular to two other vectors X and Y.
The dot product is calculated by multiplying together the first element, the second element and so on of X and Y; these products are summed to yield the dot product. The dot product is also found by multiplying together the length of X, the length of Y and the cosine of the angle between X and Y. Calculating the cross product requires finding the determinant of a matrix in which the first row contains unit vectors, the second row contains the vector X and the third row contains the vector Y. Although the dot product is commutative (X dot Y is the same quantity as Y dot X), the cross product is not. X cross Y is the determinant of the matrix described above, while Y cross X is the determinant of the matrix where the first row contains unit vectors, Y is the second row and X is the third row.