Difference Between Dot Product and Cross Product of Vectors

Dot product and cross product are two mathematical operations that are frequently used in linear algebra and vector calculus. So, what is the difference between dot product and cross product?

The resultant of dot product is a scalar quantity. And, the resultant of cross product is a vector quantity. Not to mention, one of the most significant similarities between them is that they both follow the scalar multiplication law.

What is Dot Product?

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,

∑ = summation

n = 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

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, and k are the unit vectors in x, y,and z directions.

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 the 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.

θ = 90°. A . B = 0

Applications of Dot Product

Applications of dot product include:

• Finding the Angle Between Two Vectors
• Finding Projections
• Work Done by a Force
• Testing for Orthogonality
• Finding the Magnitude of a Vector
• Determining the Direction of a Force, etc.

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.

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

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

|A| and |B| are the magnitudes of the two vectors

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.

θ = 90°

Applications of Cross Product

Applications of Cross Product include:

• Determining the Area of a Parallelogram
• Computing Torque
• Finding the Moment of Inertia
• Magnetic Field
• Calculating the Normal Vector
• Computing Angular Momentum, etc.

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I am a mechanical engineer by profession. Just because of my love for fundamental physics, I switched my career, and therefore I did my postgraduate degree in physics. Right now I am a loner (as 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 aspect of physics comes within the range of cosmology. And I love traveling, especially the Sole one.

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