Difference Between Dot Product and Cross Product

Introduction

In this article, we will first discuss Scalar and Vector quantities and thenceforth move on to the Dot Product and Cross Product.

Scalar Quantity - The quantities which have only magnitude; can be added, subtracted, multiplied, and divided just like ordinary numbers. A Scalar quantity is subject to simple arithmetic operations. For example- Mass, volume, time, distance, etc.

Vector Quantity - A vector quantity is the one which has both magnitude and direction. Vector quantities can be added, subtracted or multiplied by simple arithmetic rules. But, the arithmetic division of vector quantity is not possible at all. Examples - velocity, acceleration, force, etc. Vectors are like the basic blocks of multivariable.

Things get a little tricky while multiplying two vectors. A vector quantity is multiplied by using two basic kinds of products. The two products are Dot Product and Cross Product. In this article, we shall discuss in detail the differences between Dot Product and Cross Product.

Dot Product vs Cross Product

Both of them involve the multiplying of two vectors. A dot product of two vector quantities is also known scalar product, which indicates magnitude but no direction. A cross product of two vector quantities is called a vector product, indicating both magnitude and direction.

The elementary difference between a Dot product and a Cross product is that the consequence of a dot product is a scalar quantity whereas, in a cross product the outcome is a vector quantity. On one hand, where the dot product strictly follows the commutative law, on the contrary, the cross product does not.

Difference Between Dot Product and Cross Product in Tabular Form

Parameters of Comparison	Dot Product	Cross Product
Definition	Dot Product is the product of the magnitude of the two corresponding vectors and the cosine concerning them.	Cross Product is the product of the magnitude of the two vectors and the sine of the angle between them.
Denotation	The dot product of two vectors A and B is denoted by a dot (•).	The cross product of two vectors A and B is denoted by a cross (x).
Mathematical Equation	A•B =|A||B| cos θ	A× B =ABsin θ n
Direction presence	A dot product results in a scalar quantity; it does not have any direction.	The right-hand thumb rule gives the direction of the cross product.
Same direction rule	The dot product of two vectors in the same direction will be the maximum.	The cross product of two vectors in the same direction will be zero.
Result	The end product of a dot product is a scalar quantity.	In cross product, the resultant product is a vector quantity.
Vector rule	In dot product, if the vectors are perpendicular (⊥), their product will be zero. A • B = 0	In cross product, if the vectors are parallel (||) to each other, their product will be zero. A x B = 0
Commutative law	The dot product strictly follows the commutative law,i.e.A• B= B•A	The cross product does not follow the commutative law of multiplication, i.e. A × B ≠ B ×A
Orthogonality of vectors	When the vectors are orthogonal, the dot product will be zero, in the case when the angle θ = 90°.	When the vectors are orthogonal, the cross product will be the maximum i.e. the angle θ = 90°.

What is Dot Product?

The term 'dot product' comes from the centre dot that denotes the operation. It is also known as the inner product or the projection product. Dot Product is a product of the magnitude of vectors and the cosine of the angles between them. Algebraically, it is described as the sum of two corresponding entries of the two sequences of numbers. It is an algebraic operation that yields a single integer from two similar series of numbers. The final product is a scalar quantity. We take the two corresponding components, multiply them, and everything together.

The first thing to notice is that the dot product of two vectors gives us a number. The dot product operation is utilised to describe the length between two points on a plane, also known as coordinates. The dot product has four properties which have been explained below- following scalar multiplication, commutative, orthogonal, and distributive. Read further for more properties as well as the application of the dot product.

Properties of Dot Product

• A dot product obeys the commutative law of multiplication.

The equation is as follows-  A• B= B•A 

• It is a product of the magnitudes of two vectors taken as A and B, and the cosine angle between them.

A •B = |A| |B| cosθ

• When θ = π/2, the two vectors are perpendicular to each other. It will result in the dot product between them being 0.
• A dot product can also be negative if the two vectors point in opposite directions. 
• The dot product of two vectors aids several operations in different fields like engineering, geometry, astronomy, and mechanics.
• A dot product is associative in terms of scalar multiplication.
• It is distributive when it comes to vector addition.
• When the angle between the two vectors equals zero degrees, their dot product will equal zero.
• The dot product will be positive only if the vectors are in the same direction. The product will be as additional as possible when the corresponding vectors are in the same direction.
• The dot product of two equal vectors is orthogonal if their product is zero, which means θ = 90°
• The dot product measures the similarity of two vectors or how well they travel together. In simple words, if they are parallel (in the same direction), the dot product will be as more as possible; if the vectors are perpendicular (travelling in opposite directions), the dot product will be zero.
• Dot product does not follow the cancellation law, unlike the multiplication of ordinary numbers.

Application of Dot Product

• Dot Product is employed to estimate the work done. Work is the multiplication of the force applied and the displacement that occurred. If the force is applied at an angle θ to the displacement, the work done will be calculated as - W= f d cos θ
• It is also applied to discover whether the two vectors are orthogonal or not.
• The dot product is like a portal to multiply two vectors.
• It is calculated from the product of the two vectors because it is a scalar quantity with no direction and only magnitude. 
• The dot product operations are employed in three-dimensional space for inducing equations for representing lines, spheres, and planes.
• It aids in calculating the angle created by two vectors and perceiving the location of the vectors in the coordinate axis.

What is Cross Product?

The second way of multiplying two vectors is Cross Product. It is not easy defining this method as it has few complicated properties. But it is necessary to understand this method as it will pay off. Cross Product is a product of two linear vectors and the sine of the angle between them. The result of a cross product is always a vector quantity. This is the reason why it is also called a Vector product. It can also be described as an external product dimension except for three dimensions. The exterior product of two vectors is known as a bivector.

Having both magnitude and direction means that the magnitude is taken using the sine of the corresponding angles. The right-hand thumb rule is commonly used to find the direction. In this case, the two fingers represent the vectors and the thumb represents the product.

It also has four properties - distributive, commutative, orthogonal, and association with scalar multiplication. Some other properties are also described below.

Properties of Cross Product

• The cross product is not commutative in multiplication, unlike the dot product.

 A × B ≠ B ×A

• The result of a cross product will be another vector and not a number. For example- A × B = C
• The cross product always works in 3D because in 2D, not every time a vector is perpendicular to any pair of other vectors. 
• The outcoming vector of the cross product of two vectors will always be perpendicular.
• It is also distributive in nature, similar to the dot product:-

 A × (B + C) = (A × B) + (A × C)

• The cross product is also appropriate for the scalar multiplication law.
• The cross product is orthogonal only if θ = 90°.

Application of Cross Product

• The cross product gives us a way of calculating the minimum distance between two vectors so it can help compute the shortest distances.
• It is also used to determine whether two vectors are coplanar (in the same plane) or not.
• It is used to calculate the normal for a polygon or a triangle, performed in computer graphics.
• The cross product is utilised in determining the sign of the acute angle in the computational geometry of a plane.
• The cross product also aids in calculating the volume of tetrahedron or parallelepiped, which are the types of polyhedrons.
• The cross product is employed to express the Lorentz force that is encountered by a moving electric charge.
• The cross product is used to express the formula of the vector operator curl.

Main Differences Between Dot Product and Cross Product (In Points)

• The major difference between the two terms is that- the cross product is specified by magnitude along with direction, whereas the dot product is completely itemized by magnitude only.
• On one hand, where dot product results in a scalar quantity (consisting only of magnitude) on the other hand, the cross product gives out a vector quantity.
• A dot product is the multiplication of the magnitude of two vectors and the cosine of the angle between them. A cross product is the exponentiation of the magnitude of two vectors and the sine of the angle between them.
• The quantities which are totally specified by both magnitude as well as direction are called Vector quantities. Whereas, the quantity that is completely specified by the magnitudes only are known as scalar quantities.
• They both have different formulas.

For the Dot product, the formula will be:- A•B =|A||B| cos θ 

For the Cross Product:- A× B =ABsin θ n , where ‘n’ is the unit vector perpendicular.

• Dot product is also called the inner product or the projection product. Cross product is also known as the Directed Area product.
• In the case of a Dot Product- when the vectors are orthogonal, the dot product will be zero. In the case of a cross product - when the vectors are orthogonal, the cross product will be the highest.
• When the vectors are perpendicular to each other, the dot product is zero. When the vectors are parallel to each other, the cross product will be zero.
• In a similar way, when the vectors are in the same direction, the cross product results in zero. Whereas, when the two vectors are in the same direction the dot product is the maximum.

Conclusion

Let’s revise this a bit:

A dot product-

• is a scalar quantity
• is a product of the magnitude of two vectors
• is the cosine of the angles concerning the two vectors
• has magnitude
• has no direction
• has the mathematical formula- A•B =|A||B| cos θ

A cross product-

• is a vector quantity
• a product of the magnitudes of two vectors
• sine of the angles of the vectors
• has magnitude
• has direction as well
• formula- A× B =ABsin θ n

Hopefully, this article will clear the air about the two vital topics used in mathematics, physics, astronomy, engineering, and some others. They both have some highlighting contrast as well as some similarities. For example, they both are distributive properties and both of them can be associative with scalar multiplication. Therefore, remembering the differences along with the similarities is essential for better understanding. This article clearly explains the similarities and differences and gives a detailed description of Dot Product and Cross Product.