In this article, you will learn about the cross product.
You apply cross product in vectors. So, it becomes essential to learn about vectors first.
Vectors in mathematics refer to something very definitive. Vectors can be seen as a mathematical representation of physical quantity that indicates both magnitude and direction with the help of an arrow. Sometimes they are also divided into components to demonstrate how many vectors are there in each direction. The elements of a vector are present in three dimensions.
For example- x,y, and z are three components of vectors in three different directions. These components can be written as sub x, sub y, and sub z.
To understand what kind of quantity vectors represent, try taking a position. To determine position, there is a position vector. A position vector can tell where an object is located based on its point of origin. If you have less time and are in hurry in finding the cross product between 2 vectors then there are many cross product calculators on the web.
To understand it more, let’s take an example-
Suppose Sam’s house is 5 miles away from John’s. However, this will not tell John exactly where his house is located. John needs to know how far his place is (magnitude) and its direction to find his home. In this example, it is 40 degrees north of the east.
There are two ways to multiply vectors- dot product and cross product.
Cross product can tell you which part of the vector is perpendicular to the other.
If two vectors a and b are taken, then, in this case, their cross product will be another vector. In addition to that, if you know the magnitude of both vectors a and b, then you will be able to find the magnitude of the cross product. It can be achieved by multiplying the magnitude of a with the magnitude of b along with the sine of the angle between them. i.e
× = sinΘ
By definition- if we take and two non-zero and non-parallel vectors. Then the vector or cross product ( ×) in that order can be defined as a vector whose magnitude is sinΘ.
Where Θ is the angle between vector a and b whose direction is perpendicular to the plane of and in a way that, and this direction forms a right-handed system.
Now, we know that the cross product of two vectors always points out to the direction that is perpendicular to both the vectors and the plane where the two vectors lie.
By considering this, if I draw two vectors on paper, their cross product should point either into the paper or out of the paper. At the same time, it also depends on the arrangement of the vectors. Thus, if “a” vector rotates clockwise in the direction of “b”, then the cross product will point into the page. And if it spins counterclockwise, then it will point out of the page.
However, the order of the two vectors matters the most- × = ×
Unlike ordinary multiplication and dot product, the cross or vector product is non-commutative. So, be careful to keep your vectors in the proper order while calculating.