Dot Product vs Cross Product (Tabular Form)

The basic difference between dot product and cross product is that the resultant of the dot product is a scalar quantity. While, on the other hand, the resultant of the cross product is a vector quantity.

The two basic ways to manipulate the vector algebraic operations are the dot product and cross product. In fact, they are the most important ones.

Well, before going ahead with questions like why the dot product of two vectors is a scalar? Or, why the cross product of two vectors is a vector? Let me give you a brief review of the main difference between the dot product and cross product.

 

Dot Product vs Cross Product (Tabular Form)

  Dot Product Cross Product
1. The dot product is a product of the magnitude of the vectors and the cosine of the angle between them. The cross product is a product of the magnitude of the vectors and the sine of the angle between them.
2. Mathematically, the dot product is represented by A . B = A B Cos θ Mathematically, the cross product is represented by A × B = A B Sin θ
3. The end result of the dot product of vectors is a scalar quantity. The end result of the cross product of vectors is a vector quantity.
4. The dot product of vectors does not have any direction because it’s a scalar. The direction of the cross product of vectors is given by the right-hand rule.
5. If the vectors are perpendicular to each other then their dot product is zero i.e A . B = 0 If the vectors are parallel to each other then their cross product is zero i.e A × B = 0
6. The dot product strictly follows commutative law. The cross product does not follow commutative law.
7. The dot products are distributive over addition. The cross products are also distributive over addition.
8. They follow the scalar multiplication law. They too follow scalar multiplication law.

As from the above Dot Product vs Cross Product tabular form, you got a glimpse of these two vectors algebraic operations. Conversely, in order to get to know them in detail, let us try to understand both of them in a detailed format. Keep reading!

 

What is Dot Product?

dot-product-formula
Illustration showing how to find the angle between vectors using the dot product. Credit: Wikimedia Commons

The dot product is nothing but a product of the magnitude of the vectors and the cosine of the angle between them. The resultant of the dot product of vectors is always a scalar quantity. Hence, the resultant has only magnitude.

In order to align the vectors in the same direction, we take the cosine of the angle between vectors. As a result, the resultant of the dot product of vectors does not have any direction, hence, also known as the scalar product.

Apart from being known as a scalar product, the dot product also goes by the name of the inner product or simply the projection product.

 

Dot Product Formula

According to the dot product definition, there are two ways to write the dot product formula. Let us get to know them one by one in detail.

Algebraic definition

Suppose there are two vectors;

Where,  a = [a1, a2, a3, ….., an]

             b = [b1, b2, b3, ……, bn]

According to the algebraic definition, the vector dot product formula is:

A . B = ∑ ai . bi = a1b1 + a2b2 + a3b3 + …… + anbn

Where denotes summation and n is the dimension of the vector space.

Geometric definition

According to the geometric definition, the vector inner product or scalar product formula is:

A · Β = |A| |B| cos θ

Where A and B are Euclidean vectors and θ is the angle between vectors.

Special Mention

While calculating the vector dot product, the following set of rules should be kept in mind.

  • i . i = 1, i . j = 0, i . k = 0
  • j . i = 0, j . j = 1, j . k = 0
  • k . i = 0, k . j = 0, k . k = 1

Where i, j, k are the unit vectors in x, y,and z direction.

 

Properties of Dot Product

Apart from being scalar in nature, a dot product has the following properties:

Commutative

Dot products or vector inner products are commutative in nature.

A · Β = |A| |B| cos θ = |B| |A| cos θ = A . B

Or, simply A .B = B . A

Distributive

Dot products are distributive in nature.

Α · (B+C) = A · B + A · C

 

Scalar Multiplication Law

Dot products strictly follow scalar multiplication law.

(μA) . (νB) = μν (A . B)

 

Orthogonal

The dot product of two vectors is orthogonal, only and only if, their product is zero i.e θ = 90°.

A . B = 0

 

Applications of Dot Product

Dot products or scalar products are mainly used to define the length between two points in a plane, of course, when their coordinates are known.

 

What is Cross Product?

A vector cross product is nothing but a product of the magnitude of the vectors and the sine of the angle between them. The resultant of the cross product of vectors is always a vector quantity, that’s why also known as the vector product.

right-hand-rule
An illustration showing how to find the direction of the cross product by the right-hand rule.

Hence, the resultant has magnitude as well as direction. The resultant vector of the cross product of two vectors is always perpendicular. Therefore, the direction of the cross product of vectors can be determined by the right-hand rule.

Apart from being known as a vector product, the vector cross product also goes by the name of the directed area product.

 

Cross Product Formula

cross-product-formula
The cross product illustration with respect to a right-handed coordinate system. Credit: Wikimedia Commons

The vector cross product formula is defined as:

A × Β = |A| |B| sin θ n

Where A and B are two vectors, θ is the angle between A and B, and |A| and |B| are the magnitudes of the two vectors. And, of course, n is the unit vector perpendicular to the plane containing A and B.

Special Mention

While calculating the vector or cross product, the following set of rules should be kept in mind.

  • i × j = k
  • j × k = i
  • k × i = j

Where i, j, k are the unit vectors in x, y, and z-direction.

 

Properties of Cross Product

Apart from being vector in nature, a cross-product has the following properties:

Non-Commutative

Cross products are non-commutative in nature.

A × B ≠ B × A

 

Distributive

Just like dot products, cross products are also distributive in nature.

A × (B + C) = (A × B) + (A × C)

 

Scalar Multiplication Law

Cross products are also compatible with scalar multiplication law.

(μA) × (B) = μ (A × B)

 

Orthogonal

The cross product of two vectors is orthogonal, only and only if, their product is maximum i.e θ = 90°.

 

Applications of Cross Product

Cross products or vector products are mainly used in computational geometry such as to define the distance between two skew lines. They are also often used to determine if two vectors are coplanar or not.

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I am a mechanical engineer by profession. Just because of my love for fundamental physics, I switched my carrier, and therefore I just completed my postgraduate degree in physics. Right now I am a loner (as like ever) and a Physics blogger too. My sole future goal is to do a Ph.D. in theoretical physics, especially in the field of cosmology. Because in my view, every part of physics comes within the range of cosmology. And I love traveling, especially the Sole one.

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