Difference WikiPhysics

Difference Between Dot Product and Cross Product

Main Difference

The main difference between Dot Product and Cross Product is that Dot Product is the product of two vectors that give a scalar quantity, whereas Cross Product is the product of two vectors that give a vector quantity.

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Dot Product vs. Cross Product

The dot product is the product of two vector quantities that result in a scalar quantity. On the other side, the cross product is the product of two vectors that result in a vector quantity. The dot product is also identified as a scalar product. On the flip side, the cross product is also known as the 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 with 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, i.e., “n.” So, cross product can be defined as A × B = AB Sinθ n.

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A dot product follows commutative law (According to this law, the sum and product of two factors do not change by changing their order) as A . B =B . A. Conversely, the cross product does not follow the commutative law, i.e., A × B ≠B × A.

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

Comparison Chart

Dot ProductCross Product
The product of two vectors that give a scalar quantity is known as the dot product.The product of two vectors that give a vector quantity is known as the cross product.
Also Known As
The dot product is also known as the scalar product.Cross product is also known as a vector product.
Denoted As
If there are two vectors named “a” and “b,” then their dot product is represented as “a . b.”Cross product of two vectors is represented as

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“a × b.”

Calculations
The dot product can be obtained by multiplying the magnitude with 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, i.e., “n.”
Representation
The dot product can be represented as,

A . B = AB Cos θ.

Cross product can be represented as,

A × B= AB Sinθ n.

Commutative Law
A dot product follows commutative law so,

A. B =B . A.

The cross product does not follow the commutative law, i.e., A × B ≠B × A.
Zero Product
The scalar product of two vectors will be zero if they are perpendicular to each other, i.e., A . B= 0The vector product of two vectors will be zero if they are parallel to each other, i.e., A × B= 0
Uses
A dot product is used to calculate the length of a vector, 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.

What is Dot Product?

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

A dot product is an algebraic operation in which two vectors, i.e., quantities with both magnitude and direction, combine to give a scalar quantity that has only magnitude but not direction. This product can be found by multiplication of the magnitude of mass with the cosine or cotangent of the angles. So, it is written as: A . B = AB Cos θ

The two vector’s scalar product will be zero if they are vertical to each other, i.e., A . B= 0. Moreover, a dot product also follows the commutative law. According to this law, the sum and product of two factors do not change by changing their order, i.e. A . B = B . A

Uses

  • 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 so frequently.

What is Cross Product?

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.” So, the name cross product is given to it due to the central cross, i.e., “×,” which is used to designate this operation. On the other side, 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, i.e., quantities with both magnitude and direction combine and give a vector quantity in result too. This product can be found by multiplication of the magnitude of mass with the angle’s sine, which is then multiplied by a unit vector, i.e., “n.” So, it is written as

A × B= AB Sinθ n

The vector product of two vectors will be zero if they are parallel to each other, i.e., A×B= 0. Moreover, the cross product does not follow the commutative law, i.e., A×B ≠B×A.

Uses

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

Key Differences

  1. The product of two vectors that give a scalar quantity is known as dot product whereas, the product of two vectors that give a vector quantity is known as the cross product.
  2. The dot product is also identified as a scalar product. On the other hand, the cross product is also known as the vector product.
  3. If there are two vectors named as “a” and “b” than their dot product is represented as “a . b.” Conversely, the cross product of two vectors is represented as “a × b.”
  4. The dot product of two vectors can be found by multiplication of the magnitude of mass with the angle’s cosine. On the flip side, cross product can be obtained by multiplying the magnitude of the two vectors with the sine of the angles, which is then multiplied by a unit vector, i.e., “n.”
  5. 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.
  6. A dot product follows commutative law so, A.B =B.A. On the other side, the cross product does not follow the commutative law, i.e., A×B ≠B×A.
  7. The scalar product of two vectors will be zero if they are perpendicular to each other, i.e., A.B =0 while, the vector product of two vectors will be zero if they are parallel to each other, i.e., A×B=0.
  8. A dot product is used to calculate the length of a vector, projection of a point, or the angle between two vectors, etc. On the flip side, a cross product is used to find the specular light and a vector that is perpendicular to the plane covered by two vectors, etc.

Comparison Video

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Conclusion

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

Harlon Moss

Harlon currently works as a quality moderator and content writer for Difference Wiki. He graduated from the University of California in 2010 with a degree in Computer Science. Follow him on Twitter @HarlonMoss