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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 two vectors’ Cartesian coordinates are combined, the dot product of their Cartesian coordinates is frequently employed in Euclidean geometry. Even though it is not the only inner product that can be defined on Euclidean space, it is frequently referred to as the inner product (or, more rarely, the projection product) of Euclidean space because of its shape.
The dot product is defined as the sum of the products of the corresponding entries in the two sequences of numbers, in terms of mathematics. According to the geometrical definition, it is equal to the product of the two vectors’ Euclidean magnitudes and the cosine of the angle between them. When Cartesian coordinates are used, these definitions are equivalent to one another.
While the term “dot product” comes from the centred dot “.” which is commonly used to designate this operation, an alternate term is 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.
Difference Between Dot Product and Cross Product
Dot product | Cross product | |
1 | This product is formed by multiplying the magnitudes of the vectors by the cosine of their angle with respect to one another. | It is a product of the magnitudes of the vectors and the sine of the angle between them that is known as the cross product. |
2 | When expressed mathematically, the dot product is denoted by the symbol A. B = A B Cosθ. | When expressed mathematically, the cross product is denoted by the expression A×B = A B Sinθ. |
3 | The dot product of vectors yields a scalar quantity as the final result of the operation. | Ultimately, the cross product of vectors produces a vector quantity as the final result. |
4 | Due to the fact that it is a scalar, the dot product of vectors does not have any direction. | The right-hand rule can be used to determine the direction of the cross product of two vectors. |
5 | It is true that if two vectors are perpendicular to each other, their dot product is zero, which is A. B= 0. | The cross product of two vectors is zero if they are parallel to each other, as in A × B=0. |
6 | The dot product strictly adheres to the rules of commutative law. | In this case, commutative law does not apply to the cross product. |
7 | The dot products are distributive in nature rather than additive. | In addition, the cross products are distributive rather than additive. |
8 | They are guided by the law of scalar multiplication. | They, too, are subject to the law of scalar multiplication. |
Scalar Product
Scalar Product is a vector multiplication operation that is performed on a vector. The scalar product of two vectors is equal to the sum of the products of the corresponding components of the vectors in the two vectors’ respective directions. In other words, the scalar product is equal to the product of the magnitudes of the two vectors plus the cosine of the angle formed by the two vectors intersecting. It is a scalar quantity that is also referred to as the dot product of vectors in some circles.
It is possible to compute the scalar product of two vectors by multiplying the modulus of the first vector by the modulus of the second vector, and the cosine of the angle formed by the first and second vectors. In other words, the scalar product is the result of the magnitude of the first vector and the projection of the first vector onto the second vector, where the magnitude of the first vector is the first vector’s magnitude. Using two vectors a and b, the following is the scalar product formula:
b = |a| |b|cosθ
Conclusion
The dot product, also known as the scalar product, of two vectors is a number (Scalar quantity) that can be obtained by performing a specific operation on the vector components of the two vectors. 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. The dot product is defined as the sum of the products of the corresponding entries in the two sequences of numbers, in terms of mathematics. According to the geometrical definition, it is equal to the product of the two vectors’ Euclidean magnitudes and the cosine of the angle between them.
The dot product of vectors yields a scalar quantity as the final result of the operation.
The dot products are distributive in nature rather than additive. They are guided by the law of scalar multiplication. The dot product strictly adheres to the rules of commutative law.
A scalar product is a vector multiplication operation that is performed on a vector. It is a scalar quantity that is also referred to as the dot product of vectors in some circles.
