Dot Product Of Two Parallel Vectors

The total of the products of the matching entries of the 2 sequences of numbers is the dot product. It is the sum of the Euclidean orders of magnitude of the two vectors as well as the cosine of the angle between them from a geometric standpoint. When utilising Cartesian coordinates, these equations are equal. If and only if two vectors A and B are scalar multiples of one another, they are parallel. Vectors A and B are parallel and only if they are dot/scalar multiples of each other, where k is a non-zero constant. In this article, we’ll elaborate on the dot product of two parallel vectors. 

Dot Product

The dot product, also known as the scalar product, is an algebraic function that yields a single integer from two equivalent sequences of numbers. The dot product of a Cartesian coordinate system of two vectors is commonly used in Euclidean geometry. Although it is not the only inner product that may be written on Euclidean space, it is frequently referred to as “the” inner product.

Dot Product Characteristics

The characteristics of the dot product of vectors are as follows.

• Vector identities 

• Commutative property 

• Distributive property 

• Natural property 

• General properties

Parallel Vectors

Two parallel vectors are usually scalar multiples of one another. Assume that the two vectors, namely a and b, are described as follows: b = c a

Where c is a real-number scalar. The vector b is a scalar multiple of vector an in the previous equation, and the two vectors are seen to be parallel. The orientation of vector b is determined by the sign of scalar c.

If c is greater than zero, both vectors would have the same orientation. If c is negative, such that, if c 0, the vector b would point in the opposite direction as the vector a.

Similarly, the vector a could be written as 

a = 1/c b using the preceding equation.

As a result, for every two vectors to be parallel, they should be scalar multiples of one another. Consider the situation where the value is 0. We may then write: 

b = 0× a 

b = 0

In this scenario, the vector b would become a zero vector, as well as the zero vector is regarded parallel to all other vectors.

Dot Product Of Two Vectors

The dot product of two vectors is a quantity that describes how much force 2 distinct vectors generate in the same direction. The scalar product of the two vectors is a value that is obtained by multiplying the  parts of each vector.

A dot product is a scalar quantity that changes with the angle between two vectors. Because the fraction of a vector’s total force committed to a given direction increases or decreases depending on whether the entire vector is pointing either towards/ away from that direction, the angle between both the vectors has an effect on the dot product.

Dot Product Of Two Parallel Vectors

A scalar product  A. B of two vectors A and Bis an integer given by the equation

A. B= ABcosΘ

In which, is the angle between both the vectors  Because of the dot symbol used to represent it, the scalar product is also known as the dot product.

The direction of the angle somehow isnt important in the definition of the dot product, and it could be evaluated of either of the two vectors to another because cos = cos (- ) = cos ( 2 – ). When 90° < < 180° is a negative number, and when 0° 90° is a positive number, the dot product is a negative sign number. 

When two vectors having the same direction or are parallel to one another, the dot product of the two vectors equals the magnitude product.

Taking,,

  = 0 degree

so, cos 0 = 1

Therefore,

A. B = ABcos = AB

Conclusion

To conclude, The dot product, also known as the scalar product, is an algebraic function that yields a single integer from two equivalent sequences of numbers. The dot product of a Cartesian coordinate system of two vectors is commonly used in Euclidean geometry. Two parallel vectors are usually scalar multiples of one another. Assume that the two vectors, namely a and b, are described as follows: b = c* a, where c is a real-number scalar. When two vectors having the same direction or are parallel to one another, the dot product of the two vectors equals the magnitude product. Dot product of two parallel vectors: Taking, = 0 degree, cos 0 = 1 which leads to, A. B = ABcos = AB