Differences

Dot Product vs Cross Product (Tabular Form)

Dot Product vs Cross Product

The main difference between Dot Product vs Cross Product is that the outcome of dot product is a scalar quantity. The cross product’s vector quantity is the resultant.

The dot product and cross product are the two fundamental operations used in vector algebra. In fact, they are the most essential.

Before asking why the dot product of two vectors is a scalar, let’s address this first. Alternatively, why is the cross product of two vectors a vector? I’ll quickly go through the major distinction between the dot product and cross product.

Dot Product vs Cross Product (Tabular Form)

Try to Understand:

 Dot ProductCross 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 θMathematically, 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 = 0If 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.

You received a glimpse of these two vector algebraic operations from the tabular form above that compares the dot product vs cross product. On the other hand, let’s strive to understand each of them in detail so that we can come to know them better. Go on 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 magnitude of the vectors and the cosine of the angle between them are simply multiplied to get the dot product. A scalar quantity is always the dot product of two vectors. As a result, the outcome just has magnitude.

We take the cosine of the angle between the vectors in order to align them in the same direction. Due to the lack of direction in the dot product of vectors, it is also referred to as the scalar product.

The dot product is referred to as a scalar product as well as the inner product or simply the projection product.

Dot Product Formula

There are two methods to write the dot product formula, according to the definition of the term. Let’s learn more about each of them individually.

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

Naturally, when their coordinates are known, dot products or scalar products are typically employed to define the distance between two points in a plane.

What is Cross Product?

The sine of the angle between the vectors and their magnitude are combined to form a vector cross product. Because the outcome of a vector cross product is always a vector quantity, it is often referred to as a vector product.

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

As a result, the outcome has both magnitude and direction. When two vectors are cross-producted, the resulting vector is always perpendicular. As a result, the right-hand rule can be used to identify the direction of the cross product of vectors.

The vector cross product is also referred to as the directed area product in addition to being a vector 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 the magnitudes of the two vectors, is the angle between A and B, and A and B are two vectors. Naturally, n is the unit vector perpendicular to the plane where A and B are located.

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

The major application of cross products or vector products in computational geometry is to specify the separation between two skew lines. They are frequently used to establish whether or not two vectors are coplanar as well.

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