Vector Product: Properties and Types

In Physics, terms such as force, velocity, speed, and work are usually classified as a scalar or vector quantity. 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 vector products can be easily performed on them. Product, in particular, can be performed on scalars and vectors, i.e., cross product, scalar product, and vector product.

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, toward 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 toward 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.

Vector Triple Product

It is a subdivision of vector algebra in which the cross-product of three independent vectors is taken into consideration. To get the vector triple product, multiply the cross product of one vector by the cross products of the other two vectors and divide the result by three. The method produces a vector as a consequence of its execution. If three vector are A, B and C then the vector triple product is ABC=( A.C)B-(C.A)B. 

Vector triple product is an area of vector algebra that has a long history of research and publication in the discipline. When we look at a vector triple product, we may learn more about the cross-product of three vectors, which is quite useful.

Results of the triple product of vectors

The cross product of a vector with the cross products of the other two vectors may be used to calculate the triple product of that vector. A vector is generated as a result. The BAC – CAB identity may be found by simplifying the vector triple product.

Vector Triple Product Formula

The cosine of the angle between two vectors is used to compute the angle between the vectors. This formula is derived by taking the product of the two vectors’ separate components and dividing it by their product of magnitude. The formula for angle between the two vectors is shown below.

cosθ=a.b|a|.|b|

cosθ=a1.b1+a2.b2+a3.b3a21+a22+a23 b21+b22+b23

Conclusion

A Vector Product is a cross product. When two vectors of different types are multiplied together, it is called a cross product. In mathematics, a cross product or vector product is a consequence of multiplying two vectors and obtaining the product to be a vector quantity as well. It is perpendicular to the plane containing the two provided vectors, resulting in the final vector.