Cross product formula

About Cross product formula

The cross product is a binary operation on two vectors in three-dimensional space. It produces a new vector that is perpendicular to both. The right-hand rule is used to calculate the cross-product of two vectors.

According to the right-hand rule, the resultant of any two vectors is perpendicular to the other two vectors. We may also determine the magnitude of the resulting vector by using cross-product.

Definition of Cross Product Formula

Vector product/cross product of vectors $\overrightarrow{A}$ and $\overrightarrow{B}$ is denoted by $\overrightarrow{A}$ × $\overrightarrow{B}$ and its resultant vector is perpendicular to the vectors $\overrightarrow{A}$ and $\overrightarrow{B}$.

Cross Product Formula

Formula is given as :

$\overrightarrow{A}$ × $\overrightarrow{B}$= ABSinθ &ncirc;, where θ is the angle between the given vectors,

Here, &ncirc; denotes unit vector

Cross Product of Two Vectors

cross product of two vectors is indicated as,$\overrightarrow{X}$ × $\overrightarrow{Y}$ = |$\overrightarrow{X}$|.|$\overrightarrow{Y}$| sinθ

Let us take any two vectors $\overrightarrow{X}$ = x$\overrightarrow{i}$ + y$\overrightarrow{j}$ + z$\overrightarrow{k}$and $\overrightarrow{y}$ = a $\overrightarrow{i}$ + b$\overrightarrow{j}$ +c $\overrightarrow{k}$

Matrix form, also known as determinant form, can be used to define the cross product of these two vectors.

$\overrightarrow{X}$ × $\overrightarrow{Y}$ = $\overrightarrow{i}$(yc -zb) - $\overrightarrow{j}$(xc - za) + $\overrightarrow{k}$(xb - ya)

Cross Product Rules

Anti-Commutative Property: Cross product is anti-commutative property i.e., it means $\overrightarrow{A}$ × $\overrightarrow{B}$ = -$\overrightarrow{B}$ × $\overrightarrow{A}$

let $\overrightarrow{A}$ = a ₁$\overrightarrow{i}$ + a ²$\overrightarrow{j}$ + a ³$\overrightarrow{k}$ and $\overrightarrow{B}$ = b₁$\overrightarrow{i}$ + b₂$\overrightarrow{j}$ + b ³$\overrightarrow{k}$ ,

= -$\overrightarrow{A}$ × $\overrightarrow{B}$

Therefore, $\overrightarrow{A}$ × $\overrightarrow{B}$ = - $\overrightarrow{B}$ × $\overrightarrow{A}$

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