Introduction

Vectors are objects that have both direction and magnitude. The vector product is a single vector resulting from the two vectors. Therefore, it is perpendicular to both vectors following the right-hand thumb rule. 

For example, assuming A and B to be the two vectors and C to be the vector product, we can write A and B’s vector product (C) as C=AxB=(AB sinθ)n. Vectors can be multiplied in 2 ways, first is the Dot Product of Two Vectors and second is the Cross Product of two vectors.

Listed below are the two methods of multiplying two vectors:

Dot Product of Two Vectors

The dot product of vectors is the result of the magnitude of the two vectors and the cos angle between them. For example, if the magnitude of Vector A and B is |A||B| and the angle between the vectors is cosθ, then the dot product becomes A.B= |A||B|cosθ.

 This formula represents the magnitude and the angle together,where (A.B) is the dot product, |A| is the magnitude of vector A , |B| is the magnitude of the vector of B and cosθ is the cosine angle in the middle of the two vectors. In some scenarios, the value of the dot product might be nil.

 For example, if the value of any one vector or both vectors is 0, the angle between is not considered, the dot product equals 0. Therefore, the formula of the dot product of vectors is A.B=|A||B|cosθ.

Properties of Dot Product:

  1. The dot product is commutative. For instance, the dot product a.b can become b.a
  2. The dot product represents the perpendicularity of the two vectors. If a.b=0, then it can be clearly seen that either a or b is zero or cos θ = 0. 
  3. The dot product to vector itself is equal to the squared magnitude of the vector itself i.e. a.a = a2
  4. It follows the distributive law  i.e. a.(b + c) = a.b + a.c
  5. Scalar multiplication of vectors in dot product i.e. (pa) . (qb) = pq (a.b)
  6. i.i = j.j = k.k = 1

Cross Product of two vectors

The cross product of the two vectors is the result of the magnitude of the two vectors and the sinθ angle between both vectors. For example, if the two vectors’ value is A and B, the magnitude of the vectors is |A||B|, and the sine angle between them is sinθ. Therefore the formula of the cross products between two vectors is AxB=|A||B|Sinθ.

Right-Hand Cross Product Rule:

The right-hand cross product rule highlights the direction of the vector. This rule states that we must stretch the index finger of the right hand towards the direction of the first vector (A) and the middle finger towards the second vector (B). As a result, the hand’s thumb will show the direction of the cross product (AxB). 

Cross Product of Perpendicular Vectors

If vectors A and B are perpendiculars, then the sin angle becomes sin 90,ie.,sin 90= 1. Therefore, the cross product of the vectors become AxB= |A|.|B|x1= |A|.|B|

Cross Product of Parallel Vectors

If vectors A and B are parallel or opposite, the sin angle is sin 0.ie., sin 0=0. Therefore, the cross product of the parallel vectors is AxB=|A|.|B|x0=0

Properties of Cross Products

Listed below are the properties of cross products:

  1. Cross Product of vectors is not interchangeable. Since the values also consider direction, the vector’s value becomes negative if any of the values are interchanged. For instance, Vectors A and B, when calculated as the cross product AxB it cannot become BxA; it becomes -BxA if the values interchange. This is called Anti-commutative property.
  2. If any of the values of the vectors are 0, then the cross product of the vectors is 0. 
  3. The cross product of the vectors has a distributive property. For instance, if the value of the vectors is A and B and the cross product of the vectors is Ax(B+C)= AxB + AxC. 
  4. i × j = k and j × i = –k and j × k = i and k × j = –i and k × i = j and i × k = –j
  5. AxB=0 implies either vectors are parallel or either of them is zero.

Differences between dot product and cross product of vectors

Listed below are the differences between dot products and cross products:

  1. The dot product considers the cosine angle between the two values of the vector. Meanwhile, the cross product of the vector considers the sine angle between the two values of the vector.
  2. The dot product of the vectors is a scalar quantity. On the other hand, the cross product of the vector is a vector quantity.
  3. The dot product of the vectors is the inner or projection product, whereas the cross product of the vectors is the direct area product.
  4. The dot product of the vectors considers only the magnitude of the vectors. Meanwhile, the cross product of the vectors considers both the magnitude and direction of the values of the vectors. 
  5. The cross product of the vectors follows the right-hand rule, while the dot product of the vectors does not follow the right-hand rule. 

Conclusion

Therefore, two ways of multiplying the vectors are dot products of vectors and cross products of vectors. The vector products consider the magnitude and direction of the values of vectors. They are different from each other in some ways. The calculation of the two methods is a bit different from each other, as exhibited by the formulas. They show specific properties which are similar yet quite different from each other. Vector products of vectors help in making a perpendicular vector to the plane. It also assists in calculating the value of torque and magnetic force.