Physics

Difference between dot product and cross product

The main difference between the dot product and the cross product is that the dot product is the product of two vectors that give a scalar quantity, while the cross product is the product of two vectors that give a vector quantity.

Key differences

  • The product of two vectors that give a scalar quantity is known as the dot product, while the product of two vectors that give a vector quantity is known as the cross product.
  • If there are two vectors named “a” and “b”, their dot product is represented as “a”. second.” In contrast, the cross product of two vectors is represented as “a × b.”
  • The dot product can be denoted as A. B = AB Cos θ. On the other hand, the cross product can be represented as A × B = AB Sinθ n.
  • The dot product of two vectors will be zero if they are perpendicular to each other, that is, AB = 0, while the vector product of two vectors will be zero if they are parallel to each other, that is, A × B = 0.
  • The dot product is also identified as a dot product. On the other hand, the cross product is also known as a vector product.
  • The dot product of two vectors can be found by multiplying the magnitude of the mass by the cosine of the angle. On the other hand, the cross product can be obtained by multiplying the magnitude of the two vectors by the sine of the angles, which is then multiplied by a unit vector, that is, “n”.
  • A scalar product follows the commutative law, so AB = BA On the other hand, the cross product does not follow the commutative law, that is, A × B ≠ B × A.
  • A dot product is used to calculate the length of a vector, the projection of a point, or the angle between two vectors, etc. On the other hand, a cross product is used to find the specular light and a vector that is perpendicular to the plane covered by two vectors, etc.

Difference between dot product and cross product in Tabular Form

dot productcross product
DefinitionThe product of two vectors that give a scalar quantity is known as the scalar product or dot product.The product of two vectors that give a vector quantity is known as the cross product or vector product.
Commutative lawA dot product follows the commutative law, so A. B = BAThe cross product does not follow the commutative law, that is, A × B ≠ B × A.
Also, know asThe dot product is also known as the dot product.The cross product is also known as the vector product.
Zero productThe dot product of two vectors will be zero if they are perpendicular to each other, that is, A. B = 0The cross product of two vectors will be zero if they are parallel to each other, that is, A × B = 0
Denoted asIf there are two vectors named “a” and “b”, then their dot product is represented as “a”. second.”The cross product of two vectors is represented as “A × b”.
ApplicationsA dot product is used to calculate the length of a vector, the projection of a point, or the angle between two vectors, etc.A cross product is used to find the specular light and a vector that is perpendicular to the plane covered by two vectors, etc.
CalculationsThe scalar product can be obtained by multiplying the magnitude by the cosine of the angles.The cross product of two vectors can be obtained by multiplying the magnitude with the sine of the angles, which is then multiplied by a unit vector, that is, “n”.
RepresentationThe dot product can be represented as to B = AB Cos θ.The cross product can be represented as A × B = AB Sinθ n.

Scalar or dot product vs cross product – Overview

The scalar product is the product of two vector quantities that result in a scalar quantity. On the other hand, the cross product is the product of two vectors that result in a vector quantity. The dot product is also identified as a dot product. On the other hand, the cross product is also known as a vector product.

If there are two vectors named “a” and “b”, then their dot product is represented as “a”. b, ”which is obtained by multiplying the magnitude by the cosine of the angles. So, it can be defined as A. B = AB Cos θ. On the other hand, a cross-product is denoted as “a × b.” which can be obtained by multiplying the magnitude with the sine of the angles, which is then multiplied by a unit vector, that is, “n”. So, the cross product can be defined as A × B = AB Sinθ n.

A scalar product follows the commutative law (according to this law, the sum and the product of two factors do not change when their order changes) as A. B = BA On the contrary, the cross product does not follow the commutative law, that is, A × B ≠ B × A.

The dot product is used to find the distance from a point to a plane and to calculate the projection of a point, etc. On the other hand, a cross-product is used to calculate specular light and to calculate the distance of a point, etc.

What is the dot product?

The dot product is the product of two vectors that give a scalar quantity. It is also recognized as a dot product. If there are two vectors named “a” and “b”, then their dot product is represented as “a”. second.” So the name ‘dot product’ is given because of its central point ‘.’ used to designate this operation. On the other hand, it is also known as the dot product because this product results in a scalar quantity.

A scalar product is an algebraic operation in which two vectors, that is, quantities with magnitude and direction, are combined to give a scalar quantity that has only magnitude but no direction. This product can be found by multiplying the magnitude of the mass with the cosine or cotangent of the angles. So, it is written as: A. B = AB Cos θ

The dot product of the two vectors will be zero if they are vertical to each other, that is, A. B = 0. Furthermore, a dot product also follows the commutative law. According to this law, the sum and the product of two factors do not change when their order changes, that is, A. B = B. A

Applications

  • Generally, it is used when a vector needs to be projected onto another vector.
  • It can also be used to get the angle between two vectors or the length of a vector.
  • A dot product is used to find the projection of a point.
  • It is also used in engineering calculations very frequently.

What is the cross product?

The cross product is the product of two vectors that give a vector quantity. It is also recognized as a vector quantity. If there are two vectors named “a” and “b”, then their cross product is represented as “a × b”. Then, it is given the name of the cross product because of the central cross, that is, “×”, which is used to designate this operation. On the other hand, it is also known as a vector product because this product results in a vector quantity.

A cross product is an algebraic operation in which two vectors, that is, quantities with magnitude and direction, are combined and also result in a vector quantity. This product can be found by multiplying the magnitude of the mass with the sine of the angle, which is then multiplied by a unit vector, that is, “n”. So, it is written as

A × B = AB Sinθ n

The cross-product of two vectors will be zero if they are parallel to each other, that is, A × B = 0. Also, the cross product does not follow the commutative law, that is, A × B ≠ B × A.

Applications

  • It is used to find a vertical vector at the level spanned by two vectors.
  • A cross product is also used to find the area of ​​a parallelogram that is made up of two vectors, so that each vector provides a pair of parallel sides.
  • It is also used in engineering calculations very frequently.
  • It is also used to calculate specular light and to calculate the distance of a point, etc.

Conclution

The above discussion summarizes that the dot and cross products are two products of vectors. Scalar product or scalar product is the product in which the result of two vectors is a scalar quantity. On the other hand, the cross product or vector product is the product in which the result of two vectors is a vector quantity.

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