A Brief Note on Vector or Cross Product

Vectors are objects in mathematics that have both magnitude and direction. The size of the vector is defined by its magnitude. It is represented by a line with an arrow, where the length of the line represents the magnitude of the vector and the direction is indicated by the arrow. If two vectors have the same magnitude and direction, they are the same. This means that if we take a vector and translate it to a new position without rotating it, the vector we get at the end is the same vector we started with. 

In a three-dimensional process, a binary vector operation is used. A cross product is a type of vector multiplication that involves multiplying two vectors of different natures or types. We can multiply two or more vectors using the cross product and dot product. When two vectors are multiplied, the product is also a vector. The cross product of two vectors, also known as the vector product, is the resultant vector. The resulting vector is in the same plane as the two provided vectors.

Cross product of two matrices

 Dot-products and cross-products are the combinations of two similar things, namely a vector and another vector. The matrices and vectors in a matrices-vector product are two very different things. As a result, a matrices-vector product cannot be called a dot-product or a cross-product.

A matrices is a rectangular arrangement or table made up of rows and columns that is used to recognize a mathematical object or one of its properties. Real numbers are referred to as scalars when working with matrices. Scalar multiplication involves multiplying each entry in the matrices by the given scalar. Matrices multiplication, on the other hand, refers to the product of two matrices. If we allow a matrices to have the vectors i j, and k as entries (OK, this may not make sense, but it’s just a tool for remembering the cross product). 

Cross product of perpendicular vectors

The area of a rectangle with sides X and Y is denoted by the magnitude product, which is equal to the cross product of two vectors. The cross product formula is as follows when two vectors are perpendicular to one another:

θ=900

As we know sin 900=1 then,

XY=X.Ysin

XY=X.Ysin 900

Which is equal to the rectangle’s area

As a result, the perpendicular vectors’ cross product becomes

XY=X.Y 

Cross product of two vectors example

The method of multiplication of two vectors is the cross product of two vectors. The multiplier sign(x) between two vectors indicates a cross product. When 2 vectors are multiplied, the product is also a vector quantity, the resultant vector is referred to as the cross product of two vectors or the vector product. The resulting vector is perpendicular to the plane containing the two given vectors.

Cross product is important in many fields of science and engineering.  Very simple examples are provided below.

Example:

We apply equal and opposite forces to the two diametrically opposite ends of the tap when we turn it on. In this case, torque is employed. Torque is defined as the product of the radius and force vectors from the axis of rotation to the point of force application.

 T=RF

 Conclusion

In this article we conclude that, In three-dimensional space, cross product is a binary operation on two vectors. It produces a perpendicular vector to both vectors. a, b denotes the vector product of two vectors,  a × b. The lines a and b are perpendicular to the output vector. Cross products are vector products that have been combined together. The cross product calculates the distance between two vectors that point in opposite directions. The cross product of vectors produces a vector quantity, so the cross product of vectors is also called a vector product. Vectors are used to visualize position, displacement, velocity, and acceleration. We frequently run out of space when drawing vectors to the scale they demonstrate, so it’s important to keep track of what scale they are drawn at.