Vector algebra is an integral part of Physics and Mathematics. It simplifies calculations and helps in the analysis of a wide variety of spatial concepts. A vector is a physical quantity that has a magnitude as well as direction. Its counterpart is a scalar quantity that has only magnitude but no direction.
A dot product of two vectors is also called the scalar product. It is the product of the magnitude of the two vectors and the cosine of the angle that they form with each other.
A cross product of two vectors is also called the vector product. It is the product of the magnitude of the two vectors and the sine of the angle that they form with each other.
Comparison Table Between Dot Product and Cross Product (in Tabular Form)
| Parameter Of Comparison | Dot Product | Cross Product |
|---|---|---|
| General Definition | A dot 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 on each other. |
| Mathematical Relation | The dot product of two vectors A and B is represented as : Α.Β = ΑΒ cos θ | The cross product of two vectors A and B is represented as : Α × Β = ΑΒ sin θ |
| Resultant | The resultant of the dot 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 dot product is zero when the vectors are orthogonal ( θ = 90°). | The cross product is maximum when the vectors are orthogonal ( θ = 90°). |
| Commutativity | The dot 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 Dot Product?
A dot product or scalar 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 the inner product or the projection product.
It is represented as :
A·Β = |A| |B| cos θ
The result is a scalar quantity, so it has only magnitude but no direction.
We take the cosine of the angle for the calculation of the dot product so that the vectors align in the same direction. This way, we obtain the projection of one vector over 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
- It follows the distributive law.
Α· ( b+c) = α·b + α·c
- It follows 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 in a plane.
It is used to find the projection of a point on the plane when its coordinates are known.
What is 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 the directed area product.
It is represented as :
A×Β = |A| |B| sin θ
The result is another vector quantity. The resultant vector is perpendicular to both the vectors. Its direction can be determined using the right – hand rule.
The following rules are to be kept in mind while calculating the cross product :
- I × j = k
- J × k = i
- K × I = j
Where I, j, and k are the unit vectors in x, y, and z-direction respectively.
The cross product has the following properties :
- It is anti-commutative.
a× b = – (b × α)
- It follows the distributive law.
a × ( b+c) = α × b + α × c
- It follows 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 skew 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 relations.
The main differences between the two are :
- The dot product of two vectors is the product of their magnitudes and the cosine of the angle that they subtend on 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 relation for the dot product is : α • b = |a| |b| cos θ. On the other hand, the relation for the cross product is: α × b = |α| |b| sin θ
- The result of the dot product of two vectors is a scalar quantity, whereas the result of the cross product of two vectors is a vector quantity.
- If two vectors are orthogonal, then their dot product is zero, whereas their cross product is maximum.
- The dot product follows the commutative law, whereas the cross product is anti – commutative.
Conclusion
Vector algebra has a great utility in various mathematical subjects. Its use is very common in geometry and electromagnetics. The dot product and cross product of vectors are the basic operations in vector algebra. They have several applications. The dot product computes a scalar quantity. This quantity is generally distance or length.
The cross product computes a vector quantity. So, we get another vector in space. We can perform operations such as addition, subtraction, and multiplication on vectors. Displacement, velocity, and acceleration are common vectors in Physics.
The concept of vector evolved over 200 years ago. Since then, it has flourished owing to the contributions of many mathematicians and scientists.
References
- https://www.osapublishing.org/abstract.cfm?uri=ol-37-5-972
- https://www.maa.org/sites/default/files/pdf/upload_library/4/vol6/Dray/Dray.pdf