We explain that what is the difference between dot product and cross product with a table. Vector algebra is an integral part of physics and mathematics. Simplifies calculations and aids in the analysis of a wide variety of spatial concepts.
The difference between the dot product and the cross product of two vectors is that the result of the dot product is a scalar quantity, whereas the result of the cross product is a vector quantity. The result is a scalar quantity, so it has only magnitude but no direction.
A vector is a physical quantity that has a magnitude and a direction. Its counterpart is a scalar quantity that only has magnitude but no direction.
A vector can be manipulated by two basic operations. These operations are the dot product and the cross product, and they have big differences.
A dot product of two vectors is also called a dot product. It is the product of the magnitude of the two vectors and the cosine of the angle between them.
A cross-product of two vectors is also called a vector product. It is the product of the magnitude of the two vectors and the sine of the angle between them.
The difference between the dot product and the cross product of two vectors is that the result of the dot product is a scalar quantity, while the result of the cross product is a vector quantity.
Dot Product Vs Cross product
Comparison Point Product Parameter Cross Product
| General definition | A scalar product is the product of the magnitude of the vectors and the cos of the angle between them. | A cross product is the product of the magnitude of the vectors and the sine of the angle that they subtend to each other. |
| Mathematical relationship | The dot product of two vectors A and B is represented as: Α.Α = ΑΒ cos θ | The cross product of two vectors A and B is represented as: Α × Β = ΑΒ sin θ |
| Resulting | The resultant of the scalar product of the vectors is a scalar quantity. | The resultant of the cross product of the vectors is a vector quantity. |
| Orthogonality of vectors | The scalar product is zero when the vectors are orthogonal (θ = 90 °). | The cross product is maximum when the vectors are orthogonal (θ = 90 °). |
| Commutativity | The scalar product of two vectors follows the commutative law: A. B = B. A | The cross product of two vectors does not follow the commutative law: A × B ≠ B × A |
What is the dot product?
A dot product or dot product of two vectors is the product of their
magnitudes and the cosine of the angle subtended by one vector over the other. It is also called an indoor product or a projection product.
It is represented as:
A · Β = | A | | B | because θ
The result is a scalar quantity, so it only has magnitude but no direction.
We take the cosine of the angle to calculate the dot product so that the vectors line up in the same direction. In this way we obtain the projection of one vector onto the other.
For vectors with n dimensions, the dot product is given by:
A Β = Σ α¡b¡
The dot product has the following properties:
- It is commutative.
Α b = b α
- Follow the distributive law.
Α · (b + c) = α · b + α · c
- Follow the scalar multiplication law.
(λα) · (μb) = λμ (α · b)
The dot product has the following applications:
- It is used to find the distance between two points on a plane.
It is used to find the projection of a point on the plane when its coordinates are known.
See Also: Difference between conduction convection and radiation
What is the cross product?
A cross product or vector product of two vectors is the product of their magnitudes and the sine of the angle subtended by one over the other. It is also called a directed area product.
It is represented as:
A × Β = | A | | B | sin θ
The result is another vector quantity. The resulting vector is perpendicular to both vectors. Its direction can be determined using the right hand rule.
The following rules should be considered when calculating the cross product:
- I × j = k
- J × k = i
- K × I = j
Where I, j and k are the unit vectors in the x, y and z directions respectively.
The cross product has the following properties:
- It is anti-commutative.
a × b = – (b × α)
- Follow the distributive law.
a × (b + c) = α × b + α × c
- Follow the scalar multiplication law.
(λα) × (b) = λ (α × b)
The cross product has the following applications:
- It is used to find the distance between two oblique lines.
- It is used to determine if two vectors are coplanar.
Main differences between dot product and cross product
The dot product and the cross product allow calculations in vector algebra. They have different applications and different mathematical relationships.
The main differences between the two are:
- The scalar product of two vectors is the product of their magnitudes and the cosine of the angle that they subtend to each other. On the other hand, the cross product of two vectors is the product of their magnitudes and the sine of the angle between them.
- The relationship for the scalar product is: α • b = | to | | b | cos θ. On the other hand, the relationship for the cross product is: α × b = | α | | b | sin θ
- The result of the dot product of two vectors is a scalar quantity, while the result of the cross product of two vectors is a vector quantity.
- If two vectors are orthogonal, then their dot product is zero, while their cross product is maximum.
- The dot product follows the commutative law, while the cross product is anti-commutative.
Conclusion
Vector algebra is very useful in various mathematical subjects. Its use is very common in geometry and electromagnetics.
The dot product and the cross product of vectors are the basic operations of vector algebra. They have various applications. The dot product calculates a scalar quantity. This quantity is generally distance or length.
The cross product calculates a vector quantity. So, we get another vector in space. We can perform operations like addition, subtraction and multiplication on vectors.
Displacement, velocity, and acceleration are common vectors in Physics.
The concept of vector evolved more than 200 years ago. Since then, it has flourished thanks to the contributions of many mathematicians and scientists.
