Cross Product

The cross product of vectors V 1 ⃗ vec V _{1} and V 2 ⃗ vec V _{2} is a vector V 3 ⃗ vec V _{3} whose magnitude is given by

V 1 ⃗ ∧ V 2 ⃗ = ‖ V 1 ⃗ ‖ . ‖ V 2 ⃗ ‖ . sin ( θ ) u ⃗ vec V _{1} and vec V _{2}= ldline vec V _{1} rdline . ldline vec V _{2} rdline . sin( %theta ) vec u

θ is the angle between the two vectors  V 1 ⃗ vec V _{1} , V 2 ⃗ vec V _{2}

Analytical Expression of the Cross Product

We have two vectors V 1 ⃗ vec V _{1} and V 2 ⃗ vec V _{2} in the basis i ⃗ , j ⃗ , k ⃗ vec i , vec j , vec k , and V 3 ⃗ vec V _{3} the cross product of these two vectors and is expressed by

V 3 ⃗ = V 1 ⃗ ∧ V 2 ⃗ = ( y 1 z 2 − y 2 z 1 ) i ⃗ + ( x 1 z 2 − x 2 z 1 ) j ⃗ + ( x 1 y 2 − x 2 y 1 ) k ⃗ vec V _{3}= vec V _{1}and vec V _{2}=( y_{1} z_{2}- y_{2} z_{1} ) vec i + ( x_{1} z_{2}- x_{2} z_{1} ) vec j + ( x_{1} y_{2}- x_{2} y_{1} ) vec k

Advice

We can find the expression of cross product easily using the determinant method (3 orders)