EDUCATIONGENERAL

Define Cross product and Its properties

Learn all about Cross Prodcuct of Two vectores

The vectors are Define Cross product and Its properties quantities that have a specific direction along with their magnitude. The vectors could be mathematically multiplied by either giving a scalar product or vector Define Cross product and Its properties as a result. The vector product is generally known as the cross product, in which the vector components multiply giving a vector as a product.

Let’s learn more about the cross or vector product and its properties in the article below.

Cross Product

The vector or cross product as stated above is the resultant product as a vector when two vectors are multiplied together. The cross product as a result of two vectors combination is perpendicular to both vectors.

We can define the cross product in terms of mathematics as a vector quantity having a direction perpendicular to its component vectors while having the magnitude equal to AB sinφ.

For instance, A and B are component vectors giving C as the product (C = AxB) then we can represent the cross product as C = AB sinφ. The A and B however represent the magnitude of both the vectors A and B. For further understanding about cross product you can take help from a vector cross product calculator.

Mathematical Interpretation of Cross product

According to this definition, the cross product of two vectors that are parallel or collinear to another is calculated to be zero, because the sin of the angle between them is either 0 or 180 degrees i.e. zero.

The cross product’s magnitude relates to the area of a parallelogram, sides of which are formed by the two vectors. Accordingly, the cross product has a maximum value when the two vectors are perpendicular, however, when the two vectors are parallel, the cross product has a zero magnitude.

Another evidence for AxB = 0 if A and B are collinear is, the two collinear vectors cannot define any plane.  Moreover, as per the definition, the cross product is parallel to k, of two noncollinear vectors having i and j as the unit vectors in a coordinate plane. Similarly, the k is the unit vector being perpendicular with the plane having i and j.

Cross Product Formula

The cross product of two vectors is calculated implying the cross product formula, which gives the area calculation between the two component vectors. The magnitude of the resultant vector (which gives the parallelogram’s area) is also calculated using the same formula. The cross product formula is denoted as

A x B = AB sinθn

In the formula above, the A and B give the magnitude of the vectors A and B, while the angle between the vectors is given by θ and n is the unit vector that is perpendicular to the plane having A and B.

Properties of Cross Product

The cross product of the two component vectors depicts some properties that must be present in order for a vector’s Define Cross product and Its properties to be cross product. The key properties that a cross product have, include the following

Distributivity of the Cross Product

Just as the scalar product holds the distributive property during vector multiplication, the cross Define Cross product and Its properties is distributive. According to the distributive property of the cross product, the two given vectors A and B, then (A + B) x C = A xC + B x C.

According to this property, multiplying the sum of two vectors by another one is equal to multiplying both vectors individually by the third vector and then summing up the products.

Anti Commutativity of the Cross Product

In contrast to the scalar product in which the commutative law exists among the combined vectors, the cross product is anticommutative.  That means when we are performing algebraic operations within the formula, the correct order of the components should be taken into consideration.   The anti commutative property of  the cross Define Cross product and Its properties depicts A x B = -B x A

Area of Parallelogram

Another property of the cross product states, the resultant vector in the cross Define Cross product and Its properties gives the area of the parallelogram. While in that parallelogram the sides of the triangle i.e component vectors must be adjacent.

Cross product of collinear vectors is zero

According to this property, the cross product is calculated to be zero, if the two Define Cross product and Its properties are parallel or going straight in the same direction. In contrast, if the two vectors are perpendicular then in the case sinθ = 1 i.e. the magnitudes will multiply in this case.

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