Extension of the laws of elementary algebra to vector s.
They include addition, subtraction, and three types of multiplication. The sum of two vectors is a third vector, represented as the diagonal of the parallelogram constructed with the two original vectors as sides. When a vector is multiplied by a positive scalar (i.e., number), its magnitude is multiplied by the scalar and its direction remains unchanged (if the scalar is negative, the direction is reversed). The multiplication of a vector a by another vector b leads to the dot product, written a ⋅ b , and the cross product, written a × b . The dot product, also called the scalar product, is a scalar real number equal to the product of the lengths of vectors a (| a |) and b (| b |) and the cosine of the angle (θ) between them: a ⋅ b = | a | | b | cos θ. This equals zero if the two vectors are perpendicular (see orthogonality ). The cross product, also called the vector product, is a third vector ( c ), perpendicular to the plane of the original vectors. The magnitude of c is equal to the product of the lengths of vectors a and b and the sine of the angle (θ) between them: | c | = | a | | b | sin θ. The associative law and commutative law hold for vector addition and the dot product. The cross product is associative but not commutative.