Cross Product Formula
Cross product is a form of vector multiplication, performed between two vectors of different nature or kind. It is also known as a vector product. It further produces a vector that is perpendicular to both the multiplied vectors. In this product, multiplication between vectors is represented using the (X) sign in between. Cross product formula is a multiplication of vectors and sine of the angle formed between them. An order in which the vectors are being multiplied matters. Cross products for two vectors, \((\vec {A} × \vec {B})\) and \((\vec{B} × \vec{A})\), are not the same.
What Is Cross Product Formula?
Cross product formula between any two vectors gives the area between those vectors. The cross product is the vector product of two vectors. The cross product formula can be expressed as,
\((\vec {A × B})=AB\text{Sin}\vec{θ_n}\)
where,
 A = magnitude of vector A
 B = magnitude of vector B
 θ = angle between vectors A and B.
\(\vec{n}\) is a unit vector
If the components of the vector are given, cross product of two vectors \(\vec {A}=a\hat{i} + b\hat{j}+c\hat{k}\) and \( \vec{B}=d\hat{i} + e\hat{j}+f\hat{k}\) turns out to be:
\({\vec{A × B}} = \begin{matrix} \hat{i} & \hat{j} & \hat{k} \\ a & b & c \\ d & e & f \end{matrix}\)
\({\vec{A × B}} = \hat{i}(bfce)  \hat{j}(afcd) + \hat{k}(aebd)\)
Solved Examples Using Cross Product Formula

Example 1: Find the cross product of two vectors using cross product formula, \({\vec{A}} = \hat{i}+ 2\hat{j} + 3\hat{k}\) and \({\vec{B}} = 4\hat{i}+ 5\hat{j} + 6\hat{k}\)
Solution:
To find: Cross Product of Two Vectors
Given:
\(\vec{A} = \hat{i}+ 2\hat{j} + 3\hat{k}\)
\({\vec{B}} = 4\hat{i}+ 5\hat{j} + 6\hat{k}\)
The cross product matrix for above two vectors can be given as:
\(\vec{A} × \vec{B} = \begin{matrix} \hat{i} & \hat{j} & \hat{k} \\ 1 & 2 & 3 \\ 4 & 5 & 6 \end{matrix}\)
Using Cross Product Formula,
\(\vec{A} × \vec{B} = \hat{i}(bfce)  \hat{j}(afcd) + \hat{k}(aebd)\)
\(\vec{A} × \vec{B} = \hat{i}(2×63×5)  \hat{j}(1×63×4) + \hat{k}(1×52×4)\)
= \(\hat{i}(3)  \hat{j}(6) + \hat{k}(3)\)
= \(\hat{i}(3) + \hat{j}(6)  \hat{k}(3)\)
Answer: \(\vec{A} × \vec{B} = \hat{i}(3) + \hat{j}(6)  \hat{k}(3)\)

Example 2: A twoforce system A and B originate from a single pint, having a magnitude equal to 20N and 50N respectively. The angle between the two vectors is 30°. Find the crossproduct of the two vectors using the crossproduct formula.
Solution:
To find: Cross product of the vectors \(\vec{A}\) and \(\vec{B}\)
Given:
\(\overrightarrow{A}\) = 20N
\(\overrightarrow{B}\) = 50N
θ = 30^{º}
Using Cross Product Formula,
\(\overrightarrow{A} × \overrightarrow{B} =AB sinθ\)
= 20 × 50 × sin30°
= 1000 × 0.5
= 500N
Answer: \(\overrightarrow{A} × \overrightarrow{B} =500N\)