Cross product of Two vectors DefinitionThe cross product of two vectors is two dimensional concept.It is a vector expressing the angular relationship between thevectors.It is a vactor value as an operation of two vectors withthe same number of components (at least three).Lets say, we have two vectors, 𝑎 and 𝑏, if |𝑎| and |𝑏|represent the lengths of vectors 𝑎 and 𝑏, respectively, andif 𝜃 is the angle between these vectors.Then, The cross product of vectors 𝑎 and 𝑏 will have the followingrelationship: 𝑎 × 𝑏 = |𝑎||𝑏|sin 𝜃
Cross product of Two vectors Geometrical InterpretationGiven the characteristics of the cross product of two vectors by therelation 𝑎 × 𝑏 = |𝑎||𝑏|sin 𝜃Now, we can interpret three possible conditions: 1. 𝑎 × 𝑏 is perpendicular to both the vectors 𝑎 and 𝑏. 2. 𝑎 × 𝑏 represents the area of parallelogram determined by the these vectors as adjacent sides. 3. If 𝑎 and 𝑏 are parallel vectors then 𝑎 × 𝑏 = 0
Let 𝒂 and 𝒃 be vectors and consider the parallelogram that the two vectors make.Then ||𝒂 × 𝒃|| = Area of the Parallelogramand the direction of 𝒂 × 𝒃 is a right angle to the parallelogram that follows the righthand ruleNote:For 𝒊 × 𝒋 the magnitude is 1 and the direction is 𝒌, hence 𝒊 × 𝒋= 𝒌.
More generally,The magnitude of the product equals the areaof a parallelogram with the vectors for sides.In particular for perpendicular vectors this is arectangle and the magnitude of the product isthe product of their lengths.The cross productis anticommutative, distributive over addition
The cross product(vertical -green)changesas the anglebetween the vectors(black and red)changes