Question

define laws of vector multiplication.

Answer 1

Vector multiplication refers to one of several techniques for the multiplication of two (or more) vectors with themselves. It may concern any of the following articles:

Dot product — also known as the "scalar product", an operation that takes two vectors and returns a scalar quantity. The dot product of two vectors can be defined as the product of the magnitudes of the two vectors and the cosine of the angle between the two vectors. Alternatively, it is defined as the product of the projection of the first vector onto the second vector and the magnitude of the second vector. Thus,
A ⋅ B = ||A|| ||B|| cos θ.
Cross product — also known as the "vector product", a binary operation on two vectors that results in another vector. The cross product of two vectors in 3-space is defined as the vector perpendicular to the plane determined by the two vectors whose magnitude is the product of the magnitudes of the two vectors and the sine of the angle between the two vectors. So, if n is the unit vector perpendicular to the plane determined by vectors A and B,
A × B = ||A|| ||B|| sin θ n.
Triple products — products involving three vectors.
Multiple cross products — products involving more than three vectors

Answer 2

When two or more vectors are added together, the resulting vector is called the resultant. 5.1 Commutative law for addition: 5.2 Associative law for addition: ... In dot product, the order of the two vectors does not change the result. The associative law ofmultiplication also applies to the dot product.