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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
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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
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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.
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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
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The cross product(vertical -green)changesas the anglebetween the vectors(black and red)changes
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