Dot Product Properties of Vector

It is an algebraic operation in mathematics that takes two equal-length sequences of numbers (typically coordinate vectors) and returns a single number as a result of the operation. When Cartesian coordinates are used, these definitions are equivalent to one another. In modern geometry, vector spaces are frequently used to define Euclidean spaces, which is a result of their geometrical properties. In this case, the dot product is used to define lengths (the length of a vector is equal to the square root of the dot product of the vector by itself) and angles (the angle of a vector is equal to the square root of the dot product of the vector by itself) (the cosine of the angle of two vectors is the quotient of their dot product by the product of their lengths).

While the term “dot product” comes from dot in the centre “, which is commonly used to designate this operation, an alternate term known as the “scalar product” emphasises that the result is a scalar, rather than a vector, as is the case for the vector product in three-dimensional space.

Triple product

Dot product and cross product are two ternary operations that are used in mathematics.

The following is the definition of the scalar triple product of three vectors:

       a . ( b × c) = b . ( c × a) = c . ( a × b) 

In this case, its value is the determinant of the matrix whose columns correspond to the Cartesian coordinates of the three vectors in question. Hexagonal volume of the parallelepiped defined by the three vectors. Its three-dimensional special case of the exterior product of three vectors is isomorphic to the exterior product of three vectors.

The vector triple product is denoted by the equation, 

       a × ( b × c) = ( a . c) b – ( a . b) c

It is possible to remember this identity, also known as Lagrange’s formula, by remembering which vectors are dotted together and writing it down as “ACB minus ABC.” This formula has applications in physics, where it can be used to simplify vector calculations.

Equality of vectors

When two vectors with the same magnitude and direction are compared, they are considered to be equal vectors. Consequently, a vector does not change its original position when it is translated to a new position without changing its direction or rotating, a process known as parallel translation. Both the vectors before and after the position change are equal vectors in magnitude. Nonetheless, it would be preferable if you could recall that vectors of the same physical quantity should be compared in the same context. Example: It is possible to equate the force vector of 10 N in the positive x-axis with the velocity vector of 10 m/s in the positive x-axis in a practical manner.

Vector addition

Vector addition is governed by two laws, namely the commutative law and the associative law, respectively.

Commutative law

It makes no difference in what order two vectors are added together. This law is referred to as the parallelogram law in some circles. A parallelogram with two adjacent edges denoted by a + b and another duo of edges denoted by b + a. Both sums are equal, and the value of the parallelogram is equal to the magnitude of the diagonal of the triangle.

Associative law

The addition of three vectors is independent of the addition of the pair of vectors that came before them. (a+b)+c=a+(b+c).

Conclusion

The magnitude and direction of a vector quantity are the two most important characteristics of a vector quantity. When two vectors with the same magnitude and direction are compared, they are considered to be equal vectors. Both the vectors before and after the position change are equal vectors in magnitude. Vector addition is governed by two laws, namely the commutative law and the associative law, respectively.