Scalar and Vector Product

In physics, velocity, speed, and work are usually classified as either a scalar or vector quantity. Let’s begin by discussing vector product vs. scalar product. Scalar quantities are physical quantities with magnitude and no sense of direction. Vectors consist of both magnitude and direction. Therefore, operations such as addition and multiplication can be easily performed on them. Multiplication, in particular, can be done in two ways, i.e., cross multiplication and scalar multiplication. 

Scalar multiplication 

When two vectors are multiplied together, the result is known as scalar multiplication. The scalar multiplication of two vectors can also be described as the sum of multiplications of their corresponding components. The answer obtained will always be a scalar quantity and, as a result, a real number. 

We can understand this using an example: 

Let’s say we have two vectors, A and B. When we multiply them, the scalar multiplication obtained will be as follows: 

A.B=AB cos

Here is the angle between these two vectors.

Upon expansion, this can also be written as: 

A.B=AB cos = A(B cos )=B (A cos )

Here it is clear that Bcos will be the projection of B onto A, while Acos will be the projection of A onto B. Therefore, the scalar multiplication here can be defined as a multiplication of the magnitude of A and the component of B with A.

Formula for Scalar Multiplication 

Now that we have understood the formula for the scalar multiplication of two vectors let us look at the algebraic interpretations of the scalar multiplication. 

Algebraic formula for scalar multiplication

In algebraic terms, scalar multiplication refers to the sum of corresponding entities in a series of numbers after being added together. The dot multiplication for two vectors, a and b, are as follows: 

1. a. b = i =1naibi= a1b2 + a2b2 +……………+anbn

Here Σ is the summation while n is the dimension of the vector. 

Vector Products 

If you have two vectors, a and b, then the vector product of a and b is c. 

c = a × b

As a result, the magnitude of c = ab sin, where c is the angle between a and b and the direction of c is perpendicular to both a and b. What should the direction of these cross-direction products be now? So, we use a rule known as the “right-hand thumb rule” to determine the direction.

Vector product of two vectors

Cross product is a two-vector, three-dimensional, binary operation. It creates a vector that is perpendicular to both vectors. a b represents the vector product of two vectors, a and b. Perpendicular to both a and b, the resulting vector is the same. This kind of product is known as a cross product. The right-hand rule is used to determine the cross product of two vectors.

Dot product and cross product are methods for multiplying two or more vectors. Let’s take a closer look at each of the vector products.

Vector product formula

If θ is the point between the given two vectors Y and Z, then, at that point, the equation for the cross product of vectors is given by:

Y ×Z = |Y| |Z| sinθ

Or

Y×Z=||Y||. ||Z|| sinθn

Here,

X and Z are the two vectors

||X||,||Z|| are the extents of given vectors.

θ is the point between two vectors, and n is the unit vector opposite to the plane containing the given two vectors towards the path given by the right-hand rule.

Cross Product with Right-Hand Rule

The right-hand rule may be used to determine the direction of the unit vector. Stretching our right hand in this manner, we can make sure that our index finger is pointing at our first vector and our middle finger points towards our second. This is done by pointing with one’s thumb at one’s right hand’s nth finger. The right-hand rule makes it simple to demonstrate that the cross product of two vectors is not commutative. 

Vector product of unit vectors

The three unit vectors arei , j and k. So,

1. i × i = 0
2. i × j = k
3. i ×k =j
4. j ×i = –k
5. j ×j = 0
6. j ×k = i
7. k× i=j
8. k× j= -i
9. k× k= 0

This is how we determine the vector product formula of unit vectors.

Conclusion

The dot product, also known as the scalar product, is the product of the Euclidean magnitudes of the two vectors and the cosine of the angle between them. Cross product is a two-vector, three-dimensional, binary operation. It creates a vector that is perpendicular to both vectors. In mathematics, a cross product or vector product is a consequence of multiplying two vectors and obtaining the product as a vector quantity.